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+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 5, 2023
+Letter to the Editor
+A kinematically-detected planet candidate in a transition disk
+J. Stadler1, 2 , M. Benisty1, 2, A. Izquierdo3, 4, S. Facchini5, R. Teague6, N. Kurtovic7, P. Pinilla8, J. Bae9,
+M. Ansdell10, R. Loomis11, S. Mayama12, L. M. Perez13, L. Testi3
+(Affiliations can be found after the references)
+Received 7 November 2022 / Accepted 2 January 2023
+ABSTRACT
+Context. Transition disks are protoplanetary disks with inner cavities possibly cleared by massive companions. They are prime targets to observe
+at high resolution to map their velocity structure and probe companion-disk interactions.
+Aims. We present Atacama Large (sub-)Millimeter Array (ALMA) Band 6 dust and gas observations of the transition disk around
+RXJ1604.3–2130 A, known to feature nearly symmetric shadows in scattered light, and aim to search for non-Keplerian features.
+Methods. We study the 12CO line channel maps and moment maps of the line of sight velocity and peak intensity. We fit a Keplerian model of the
+channel-by-channel emission to study line profile differences, and produce deprojected radial profiles for all velocity components.
+Results. The 12CO emission is detected out to R =∼1.8′′ (265 au). It shows a cavity inwards of 0.39′′ (∼56 au) and within the dust continuum
+ring (at ∼0.56′′, i.e., 81 au). Azimuthal brightness variations in the 12CO line and dust continuum are broadly aligned with the shadows detected in
+scattered light observations. We find a strong localized non-Keplerian feature towards the west within the continuum ring (at R = 41 ± 10 au and
+PA = 280 ± 2◦). It accounts for ∆vφ/vkep ∼ 0.4, or ∆vz/vkep ∼ 0.04, if the perturbation is in the rotational or vertical direction. A tightly wound
+spiral is also detected and extending over 300◦ in azimuth, possibly connected to the localized non-Keplerian feature. Finally, a bending of the
+iso-velocity contours within the gas cavity indicates a highly perturbed inner region, possibly related to the presence of a misaligned inner disk.
+Conclusions. While broadly aligned with the scattered light shadows, the localized non-Keplerian feature cannot be solely due to changes in
+temperature. Instead, we interpret the kinematical feature as tracing a massive companion located at the edge of the dust continuum ring. We
+speculate that the spiral is caused by buoyancy resonances driven by planet-disk-interactions. However, this potential planet at ∼41 au cannot
+explain the gas-depleted cavity, the low accretion rate and the misaligned inner disk, suggesting the presence of another companion closer-in.
+Key words. planet formation – circumstellar disks
+1. Introduction
+Planet formation appears to be a robust and efficient process, oc-
+curring both around single and multiple stellar systems (Kostov
+et al. 2016) in protoplanetary disks. The advent of high resolu-
+tion imaging facilities demonstrated that nearly all bright and
+extended disks show substructures, in particular in the small
+(micron-sized) and large (mm-sized) dust tracers seen through
+scattered and thermal light, respectively (e.g., Andrews et al.
+2018; Long et al. 2018; Rich et al. 2022; Benisty et al. 2022;
+Bae et al. 2022a). Such high resolution studies applied to the
+gas tracers allow to probe overall physical conditions in the disk,
+such as its temperature structure, its surface height (Rich et al.
+2021; Law et al. 2021), and pressure variations (Teague et al.
+2018a,b; Rosotti et al. 2020). Studies of the disk density and the
+velocity structure reveal a great complexity, including localized
+non-Keplerian features that can be attributed to embedded mas-
+sive protoplanets (e.g., Pinte et al. 2022; Wölfer et al. 2022).
+Such perturbations from smooth density and velocity distribu-
+tions can directly constrain planet formation, as it is expected
+to leave clear signatures on the disk structure (e.g., Perez et al.
+2015; Yun et al. 2019). For example, the mapping of spiral wakes
+(Calcino et al. 2022), the detection of so-called ’Doppler flips’
+(change of sign in the non-Keplerian feature; e.g., Casassus &
+Pérez 2019; Norfolk et al. 2022), of meridional flows within
+dust-depleted gaps (Teague et al. 2019a), as well as of a veloc-
+ity perturbation associated with a circumplanetary disk candi-
+date (Bae et al. 2022b) enable to zoom onto the processes of
+planet-disk interaction. While most localized kinematical per-
+turbations are analyzed empirically, statistical methods to quan-
+tify their significance have been developed and led to the de-
+tection of localized signatures possibly associated with unseen
+planets (Izquierdo et al. 2021, 2022). Prime targets to search for
+protoplanets still embedded in their birth environment are the
+so-called transition disks. As in PDS70 (Keppler et al. 2019) or
+AB Aur (Tang et al. 2017), these disks host a dust-depleted cav-
+ity that has possibly been cleared by massive companions (Zhu
+et al. 2011).
+In this Letter, we focus on RXJ1604.3-2130 A (d=144.6 pc,
+1.24 M⊙, Gaia Collaboration et al. 2022; Manara et al. 2020, re-
+spectively), hereafter J1604, one of the brightest protoplanetary
+disks of the Upper Scorpius Association in the millimeter (mm)
+regime (Barenfeld et al. 2016), that exhibits a prominent cav-
+ity in the dust continuum and CO line emission (Zhang et al.
+2014; Dong et al. 2017; van der Marel et al. 2021). J1604 has
+a stellar companion located at ∼2300 au, itself a binary with
+separation 13 au (Köhler et al. 2000). The outer disk of J1604
+was resolved with the Atacama Large (sub-)Millimeter Array
+(ALMA) (Mayama et al. 2018) and the Spectro-Polarimetric
+High-contrast Exoplanet REsearch instrument (SPHERE) on the
+Very Large Telescope (VLT) (Pinilla et al. 2015), indicating a
+nearly face-on geometry. Complementary observations are in-
+dicative of a misaligned inner disk with respect to the outer disk.
+Its variable light curve is that of an irregular dipper (Ansdell et al.
+2020), infrared scattered light observations show the presence of
+two shadows with variable morphology on timescales possibly
+shorter than a day (Pinilla et al. 2018), and ALMA 12CO (J=3–
+2) line observations show deviations from Keplerian rotation in
+Article number, page 1 of 13
+arXiv:2301.01684v1  [astro-ph.EP]  4 Jan 2023
+
+A&A proofs: manuscript no. main
+Fig. 1: ALMA observations of J1604. Panel (a) 231 GHz dust continuum, black solid contours drawn at [5, 15, 25, 35, 45]σ, the image is plotted
+with a power-law scaling of γ = 0.6. (b)12CO peak brightness temperature map computed from I0 using the Planck law with black solid contours
+drawn at [5, 10, 20, 40, 60, 65, 70] σ, pixel below 5σ are masked. (c) Peak intensity residuals after subtracting an azimuthally-averaged radial
+profile from the data, where we adjusted the colour scale such that residuals smaller than 1σ are white. The beam sizes are shown in the lower left
+corner and the position of the star is marked by a green cross. In (b) & (c), we overlaid the continuum contours in white and black, respectively.
+the cavity (Mayama et al. 2018). The position of the scattered
+light shadows are suggestive of a large misalignment (∼70-90◦).
+Measurements of the projected rotational velocity (v sini) indi-
+cate that the star is aligned with the inner disk, thus misaligned
+with the outer disk (Sicilia-Aguilar et al. 2020).
+In this work, we present new ALMA observations of J1604
+and focus on the kinematics of the 12CO (J=2–1) line. In the
+following, Sect. 2 presents the observations, Sect. 3 and 4 our
+methodology and results, respectively. Section 5 provides a dis-
+cussion of the results and Sect. 6, our conclusions.
+2. Observations, calibration and imaging
+We present new ALMA Band 6 observations (2018.1.01255.S;
+PI: Benisty) with five executions spread over two years obtained
+on 2019 April 4, July 30 and 31st, and 2021 April 29, Septem-
+ber 27. The spectral set-up was designed for continuum detec-
+tion, but includes the 12CO J=2-1 line. The data were combined
+with archival data from program 2015.1.00964.S (PI Oberg; see
+Tab. A.1). The data calibration and imaging were performed
+following the procedure of Andrews et al. (2018), with CASA
+v.5.6.1 (McMullin et al. 2007), and is detailed in Appendix
+A. The synthesized beam of the 12CO line and dust continuum
+images are 0.18′′ x 0.15′′ (102◦) and 0.060′′ x 0.039′′ (- 78◦), re-
+spectively. The rms in a line-free channel was measured to be
+1.1 mJy beam−1 (4.3 K) for CO and 10 µJy beam−1 for the dust
+continuum. Figure 1 shows the dust continuum map (left), that
+displays a cavity and a bright dust ring peaking at R ∼0.56′′
+(∼81 au), and the 12CO peak brightness temperature map (cen-
+ter) that indicates a smaller cavity in gas, with a peak at R ∼0.39′′
+(∼56 au). A selection of channel maps can be found in Fig. A.1.
+3. Methodology
+Channel maps model.
+To model the disk line intensity and
+kinematics, we use the discminer package of Izquierdo et al.
+(2021). The code uses parametric prescriptions for the line peak
+intensity, line width, rotational velocity and disk emission height
+to produce channel maps and emcee (Foreman-Mackey et al.
+2013) to maximise a χ2 log-likelihood function of the difference
+between the model and input intensity for each pixel in a channel
+map. To prescribe the model intensity, we use a generalized bell
+kernel, function of the disk cylindrical coordinates (R, z):
+Im(R, z; vch) = Ip(R, z)
+�
+1 +
+�����
+vch − v
+Lw(R, z)
+�����
+2Ls�−1
+,
+(1)
+where Ip is the peak intensity, Lw is half the line width at half
+power, hereafter ’the line width’, and Ls the line slope. vch is the
+channel velocity at which the intensity is computed and v the
+observed Keplerian line-of-sight velocity. As the disk is nearly
+face-on, the code is unable to infer an emission height, and we
+therefore assume a flat emission surface. We additionally fix the
+inclination i of the disk to the one inferred from the dust con-
+tinuum (i = 6.0◦; Dong et al. 2017) to break the degeneracy of
+M⋆ · sin i. The fitting procedure and the MCMC search are ex-
+plained in detail in appendix B, where the functional form of
+each model parameter together with its best-fit parameters are
+summarized in Table B.1. We compare selected channel maps to
+best-fit model using these parameters in Fig. A.1.
+Moment
+maps.
+The moment maps are computed with
+bettermoments (Teague & Foreman-Mackey 2018). Since the
+12CO line emission is optically thick, we fitted the following line
+profile to both data and model channel maps:
+I(v) = I0 · 1 − exp (−τ (v))
+1 − exp(−τ0)
+with τ = τ0 exp
+�−(v − v0)2
+∆V2
+�
+,
+(2)
+where I0 is the peak intensity of the line and the optical depth
+τ (v) varies like a Gaussian with v0 the line centroid, τ0 the peak
+optical depth and ∆V the width of the line (where the full width
+at half maximum FWHM = 2
+√
+ln2∆V), as used in Teague et al.
+(2022). In Fig. 1 (b), we show I0 for 12CO in units of brightness
+temperature. The corresponding v0-maps for the data and model
+are displayed in Fig. 2 (a) & (b), respectively. The moment maps
+Article number, page 2 of 13
+
+Peak Brigthness Temp. (K)
+Residual (lo - <lo>) (K)
+I (mJy/beam)
+0.2
+-15
+-10
+-5 
+0.01 0.05 0.1
+0.4
+0
+5
+10
+20
+30
+40
+50
+60
+70
+10
+15
+0
+2.0
+12CO (2-1)
+231 GHz Continuum
+(a)
+(b)
+(c)
+1.5
+1.0
+(arcsec)
+0.5
+0.0
+ffset
+-0.5
+-1.0
+-1.5
+-2.0.
+5
+2
+0
+-2
+2
+0
+-2 
+2
+0
+-2
+1
+Offset (arcsec)
+Offset (arcsec)
+Offset (arcsec)Stadler et al.: A kinematically-detected planet candidate in a transition disk
+Fig. 2: Line of sight velocity maps for data v0 (a) and discminer model vmod (b). (c) Velocity residual map after subtracting v0-vmod, where the
+dust continuum is overlaid in solid contours with equal levels as in Fig. 1. The innermost region was masked during the fit by one beam size in
+radius, shown as the grey shaded ellipse. The insets in subplots (a) & (c) zoom into the innermost region of the disk to highlight the non-Keplerian
+velocities. Contours are drawn at vsys = (4.62 ± 0.60) km s−1 in steps of 0.1 km s−1 and from -60 to 60 m s−1 in steps of 10 m s−1 , respectively. All
+maps show the synthesized beam for CO (black) and the continuum (white) in the lower left corner and are masked where the CO peak intensity
+falls below a 5σ level for panel (a) and where R > Rout for the rest.
+for ∆V and τ0, as well the error of the line centroid fitting δv0
+can be found in Fig. A.4.
+4. Results
+4.1. Dust and gas radial and azimuthal brightness profiles
+Figure 1 shows the 1.3 mm dust continuum together with the
+peak brightness temperature map I0 of the 12CO (J=2-1) line
+emission. Both dust and gas tracers show a cavity, and the 12CO
+(J=2-1) line emission extends inward of the dust continuum as
+expected if the continuum ring results from dust trapping (e.g.,
+Facchini et al. 2018b, see Fig. A.2). We note that the 12CO cavity
+appears asymmetric with respect to the position of the star and
+that we observe a gap-like feature in the 12CO peak intensity map
+at R ∼1.2′′, apparent as a plateau of I0 ≈ 31 mJy beam−1 stretch-
+ing over ∆R ≈ 0.1′′ (Fig. A.2). Interestingly, the disk shows sig-
+nificant azimuthal intensity variations (34% at R=0.56′′; 19% at
+0.39′′ for continuum and gas, respectively, see also Fig. A.3).
+Figure 1 (c) shows the residuals obtained after subtracting an
+azimuthally-averaged radial profile from the 12CO peak bright-
+ness temperature map. Azimuthal variations are clearly apparent
+within the dust cavity, with residual values of > 10σ. The fainter
+regions, distributed along the east-west direction, are broadly
+aligned with fainter regions seen in the continuum (see contours
+of Fig. 1, b and Fig. A.3) and with the shadows reported in scat-
+tered light (Pinilla et al. 2018, ; Kurtovic et al. in prep).
+4.2. Kinematical features
+4.2.1. Localized velocity residuals
+The centroid residual map in Fig. 2 (c) shows a prominent, local-
+ized non-Keplerian velocity feature of δv ≈ 60 m s−1 , between
+∼0.35′′ and 0.55′′ (i.e., 50-80 au), that is, at the edge of the dust
+continuum ring, and oriented at PA ≈ (270 ± 15)◦. To assess its
+significance we follow the Variance Peak method from Izquierdo
+et al. (2021). First, the centroid velocity residuals are folded and
+Fig. 3: Folded velocity residuals (left) and detected clusters of peak
+velocities (right) in the disk reference frame. Green wedges in the right
+plot mark the significant clusters. The position of the localized velocity
+perturbation inferred from these clusters is marked with a magenta point
+with error bars. The gray region (one beam size in radius) indicates the
+masked area, and the black dashed lines, the FWHM of the dust ring.
+subtracted along the disk minor axis to remove axisymmetric
+features. Second, a 2D scan is performed to search for peak
+velocity residuals and obtain their locations in the folded map.
+Using these detected points, a K-means clustering algorithm
+searches for coherent velocity perturbations within predefined
+radial and azimuthal bins (MacQueen 1967; Pedregosa et al.
+2011). We considered seven radial and ten azimuthal bins, which
+corresponds to a width of roughly one beam size, to identify
+clusters. The algorithm now subdivides the input residual points
+such that the center of each cluster is the closest to all points
+in the cluster, by iteratively minimizing the sum of squared dis-
+tances from the data points to the center of the cluster. This leads
+Article number, page 3 of 13
+
+Vo (km/s)
+Vmodel (km/s)
+Vo - Vmodel (m/s)
+4.2
+4.4
+4.6
+4.8
+5.0
+5.2
+4.2
+4.4
+4.6
+4.8
+5.0
+-60
+-40
+-20
+5.2
+0
+20
+40
+60
+2.0 FT
+(a)
+(b)
+(c)
+0.5
+0.25
+1.5
+0.00
+0.0
+1.0
+-0.25
+-0.5 
+(arcsec)
+0.5
+0.25
+0.000.25
+0.5
+0.0
+0.5
+0.0
+Offset
+-0.5
+-1.0
+-1.5
+-2.0 E
+2n
+6
+2n
+2
+-2
+2
+0
+-2 
+2
+0
+0
+-2
+-1
+-1
+Offset (arcsec)
+Offset (arcsec)
+Offset (arcsec)96
+0A&A proofs: manuscript no. main
+to irregularly spaced bin boundaries, since the cluster centers are
+near to the densest accumulations of points.
+In Figure 3, we show the folded velocity residual map to-
+gether with the detected peak velocity residuals (grey points).
+The location of the detected peak velocity residuals in azimuth
+and radius, within identified clusters, can be found in Fig. B.1.
+Clusters with high significance (those with peak velocity resid-
+uals larger than 3 times the variance in other clusters) are lo-
+cated within one radial and azimuthal bin shown in Fig. 3 as
+green shaded annuli and wedges, respectively. Taking the centers
+of the selected clusters allows to identify a localized perturba-
+tion at 0.28′′ ± 0.07′′ (R = 41 ± 10 au) and PA = 280◦ ± 2◦. The
+reported errors are the standard deviation of the peak residual
+point (R, φ)-locations within the selected clusters. The detec-
+tion yields a cluster significance of 5.4 σφ in azimuth and 5.3 σR
+in radius, where σ represents the standard deviation of back-
+ground cluster variances with a mean of σφ = 0.034 km2s−2 and
+σR = 0.018 km2s−2 (see black crosses in Fig. B.1). We note that a
+localized signature is robustly detected regardless of the amount
+of clusters defined, which we tested using 7-12 azimuthal or 5-
+9 radial clusters. We reported the clusters associated with the
+highest significance. Additionally, we note that there are other
+detections with lower significance at 0.65′′ (94 au). This means
+that the radial extent of the prominent perturbation is roughly
+0.40′′ (58 au) and the global peak of the folded velocity resid-
+uals is at 0.39′′ (56 au) (as can be seen in the middle panel of
+Fig. B.1). This analysis confirms the presence of a significant lo-
+calized non-Keplerian feature as identified visually in Fig. 2 (c),
+within the continuum ring.
+4.2.2. Spiral feature
+Figure 2 (c) also shows an extended arc-like positive resid-
+ual feature, beyond the dust continuum emission and cover-
+ing nearly 300◦ in azimuth, more evident in the polar depro-
+jection of the velocity residual map (Fig. 4). To assess if this
+feature is a coherent structure, we use the FilFinder pack-
+age (Koch & Rosolowsky 2015) implemented in discminer
+between 0.30′′ and 1.25′′ (43-180 au) (see Fig. A.5). As indi-
+cated by the coherent filaments, the strong localized positive ve-
+locity residual discussed in Sect. 4.2.1 seems to be the starting
+point of a spiral tracing outwards up to roughly 1.1′′ (159 au).
+In Fig. 4, we overlay an Archimedean (linear) spiral, prescribed
+by rspiral = a + b φspiral, using {a, b} = {0.48, 0.12}. Computing
+the pitch angles tan(β) = −(dr/dφ)/r, we obtain values ranging
+from 14◦ to 6◦ over the spiral extent.
+4.2.3. A possible warp in the 12CO cavity
+An additional feature clearly evident from the velocity maps is
+the highly perturbed inner disk regions. As seen in the inset of
+the v0 line centroid map in Fig. 2 (a), the iso-velocity lines show
+strong bending in the inner region (∼3 beam-sizes in diameter),
+indicative of non-Keplerian velocities. This is likely tracing a
+warp or a misaligned inner disk, as reported in Mayama et al.
+(2018); Pinilla et al. (2018); Sicilia-Aguilar et al. (2020); Ans-
+dell et al. (2020) to explain the scattered light shadows and vari-
+able photometry of J1604. Higher angular resolution deep gas
+observations are however needed to assess its morphology and
+kinematics.
+Fig. 4: Polar projection of the velocity residual map. The black solid
+line shows a linear spiral trace. The grey region indicates the masked
+area and the black dashed lines, the FWHM of the dust ring. The y-axis
+extends further than 360◦ to enhance the visibility of the spiral.
+4.3. Deprojected velocity components
+To understand the contributions from vφ, vr, vz, we produce three
+centroid residual maps for each velocity component, after de-
+projection (Eq. C.1) assuming that all the velocities are either
+azimuthal, radial or vertical (see Fig. C.2; Teague et al. 2022).
+The localized residual feature at the edge of the dust ring ap-
+pears to trace variations in the vertical vz or in the rotational
+vφ motions, or a combination of both. Radial perturbations can
+be ruled out, since it is located close to the disk red-shifted
+major axis (PAdisk=258◦), where vr,proj ≈ 0. Assuming purely
+rotational velocities, it corresponds to perturbations as high as
+δvφ ≈ 600 m s−1 (∼ 0.4 · vkepler), due to the low disk inclination.
+As seen in Fig. 2 (c), the spiral-like velocity residual feature
+does not change sign around the disk major/minor axes, which
+would occur for rotational vφ or radial vr velocity perturbations,
+respectively (see Eq. C.1). We are likely seeing vertical pertur-
+bations, which we are most sensitive to in a nearly face on disk.
+Figure C.1 shows the deprojected and azimuthally aver-
+aged radial profiles of each velocity component determined
+with eddy. For vφ, we observe super-Keplerian rotation from
+R∼0.35′′-0.70′′ (51-101 au), peaking at 0.45′′ (65 au) right be-
+yond the dust continuum. The rotational velocities then sharply
+drop to being sub-Keplerian in the inner disk regions. However,
+we stress that the azimuthally averaged velocities at the radial
+location of the strong localized perturbation (R∼ 0.3 − 0.6′′,43-
+87 au) are likely affected by the feature. We tentatively observe
+radial inflow inward of the CO intensity peak but with very large
+uncertainties on vr. Finally, we mostly detect downward vertical
+motion of the disk within R∼1.25′′ (181 au).
+5. Discussion
+5.1. Origin of v0 residuals
+In this paper, we report the detection of two main non-Keplerian
+features, in addition to highly perturbed gas velocities in the gas
+cavity, that are: (1) a localized positive residual near the edge
+of the dust ring, and (2) an extended spiral-like feature, possibly
+starting from (1). A variety of velocity residual features were
+detected in other systems, with a diverse range of inclinations
+(e.g., Wölfer et al. 2022). In the case of TW Hya, a similarly
+face-on disk, the detected perturbations are ∼40 m s−1 (Teague
+et al. 2022), lower than what is derived for (1) that can account
+Article number, page 4 of 13
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+270
+major beam size
+60
+dust ring FWHM
+50
+linear spiral
+40
+180
+30
+Postion Angle (degree)
+20
+90
+10
+0
+-10
+0
+-20
+ disk rotation
+-30
+270
+-40
+-50
+60
+180
+:
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+Radius (arcsec)Stadler et al.: A kinematically-detected planet candidate in a transition disk
+for 40% of the local Keplerian velocity assuming that the per-
+turbation is purely due to rotational velocities. This is also larger
+in the magnitude of deviation than the Doppler Flip reported in
+the HD 100546 transition disk (Casassus et al. 2022). These ve-
+locity residuals are often interpreted as tracing planet-disk inter-
+actions from massive companions (Pinte et al. 2022) that poten-
+tially carve out gaps. It is thus worth noting, that our inferred
+planet location (R = 41 ± 10 au) is close to the gap location in
+13CO (J=2-1) at 37 au reported in van der Marel et al. (2021).
+Comparison with simulations (Rabago & Zhu 2021; Izquierdo
+et al. 2022) or semi-analytical prescriptions (Bollati et al. 2021)
+allows to estimate a possible planet mass from the velocity de-
+viations. To this end, we consider Eq. 14 from Yun et al. (2019)
+that relates the change in rotational velocity δvφ to the planet
+mass Mp through two-dimensional hydrodynamic simulations.
+Since δV is the difference between the super- and sub-
+Keplerian peak, we consider the peak of the super-Keplerian
+motion (vφ/vkep)/vkep (see Fig. C.1) as a lower limit for the
+dimensionless amplitude of the perturbed rotational velocity
+(δmin
+V =0.06), and its double (δmax
+V
+= 0.12) as a upper limit. As-
+suming (H/R)p = 0.1 at the planet location (R = 41 au), we
+estimate a planet mass to roughly be between Mp ≈ (1.6 −
+2.9)Mjup (α/10−3)0.5.
+The extended spiral-like feature appears to be related to the
+significant localized velocity residual. Due to its low pitch an-
+gles and the consistent positive velocity residuals, we speculate
+that the spiral is caused by buoyancy resonances excited by a
+massive planet located within the dust ring. Indeed, in contrast
+with Lindblad spirals, buoyancy spirals are shown to exhibit
+a tightly wound morphology with predominantly vertical mo-
+tions (Bae et al. 2021). Such a spiral has also been suggested in
+TW Hya (Teague et al. 2019b), which is similar in its radial ex-
+tent as the one reported here. Interestingly, Wölfer et al. (2022)
+reports a tentative arc-feature in J1604, at R ∼ 1.0′′ ranging from
+PA≈ 160 − 200◦, probed by the kinematics of the 12CO (J=3-2)
+line emission, and partly coinciding with the spiral-like feature
+that we detect. Additional observations in optically thin tracers,
+would be very useful to assess its nature.
+5.2. Kinematic perturbations due to shadows
+The localized residual feature seems to roughly align with the
+shadows detected in scattered light. Comparing Figs. 1 (c) and
+2 (c), positive and negative v0 residuals broadly align with cold
+and hot regions in the brightness temperature of 12CO, respec-
+tively (see also Fig. A.6). In particular, the orientation of the sig-
+nificant localized velocity perturbation coincides with the west-
+ern shadow. Such a shadow can cool down the disk material and
+possibly induce a local drop in pressure support and therefore
+impact the gas velocity. In this section, we estimate whether
+the detected velocity perturbations could be caused by azimuthal
+variations in temperature. We relate the azimuthal change in tem-
+perature ∆Tφ to variations in rotational velocity ∆vφ by solving
+for the Navier-Stokes-equation in cylindrical coordinates. Fol-
+lowing the derivation in the appendix D, we obtain
+∆vφ
+vkep
+≈
+�H
+R
+�2 ∆Tφ
+T ,
+(3)
+with H/R the disk aspect ratio, T the 12CO brightness temper-
+ature tracing the gas temperature (as 12CO is optically thick)
+and vkep the Keplerian velocity. Evaluating this equation for a
+large H/R = 0.2 along a radially averaged annulus centered
+at R = (0.39 ± 0.07)′′, where we experience the strongest az-
+imuthal changes in the 12CO brightness temperature of up to
+∆T ≈ 15K over T = 60K, we estimate the change in azimuthal
+velocity to be a mere δvφ/vkep ≈ 1%. Hence, the shadows cannot
+be solely responsible for the localized velocity residual feature.
+In addition, as the shadows are nearly symmetric, we would ex-
+pect a similar feature opposite along the east direction.
+Montesinos et al. (2016) investigated the development of spi-
+rals due to pressure gradients caused by temperature differences
+between obscured and illuminated regions. In their simulations
+symmetric shadows always form two-armed spirals, however,
+they only develop for massive (0.25 M⋆) and/or strongly illu-
+minated disks (100 L⊙) which does not seem to be the case for
+J1604 (∼ 0.02 M⊙ & 0.7 L⊙, Manara et al. 2020), where we also
+only observe one spiral. Additionally, we would also expect such
+spiral features to appear in the brightness temperature residuals
+(Fig. 1 c). It is therefore unlikely that shadows are responsible
+for the extended spiral-like velocity residual feature.
+5.3. A warped / misaligned inner disk?
+The bending of the iso-velocity curves that we observe in the in-
+set of Fig. 2 (a), is reminiscent of a warped or misaligned inner
+disk (Juhász & Facchini 2017; Facchini et al. 2018a). However,
+as the inner disk is unresolved in our observations, the warp mor-
+phology can’t be derived. We note that radial inflows are also de-
+generate in appearance with warps as shown by Rosenfeld et al.
+(2014), and that our observations can not be conclusive on the
+origin of the disturbed kinematics in the innermost disk. We at-
+tempted to infer varying position angle or inclination with radius
+by fitting the innermost disk only (R≤0.5′′) with eddy and con-
+sidering a fixed stellar mass but did not find to any significant
+variations compared to our best-fit values. We therefore con-
+strain the warp to be confined within one beam size (∼0.18′′, i.e.,
+26 au) from the center. We note that we obtain a 5 % higher dy-
+namical mass of the system when fitting for M⋆ while masking
+the innermost beam size in radius, an effect predicted by hydro-
+dynamical simulations of warps (Young et al. 2022).
+While our observations cannot provide a full picture of the
+system due to a limited angular resolution, the very low mass ac-
+cretion rate and near infrared excess (Sicilia-Aguilar et al. 2020),
+as well as the gas cavity in 12CO with non-Keplerian veloci-
+ties suggest the presence of an additional, very massive (pos-
+sibly stellar) companion within the inner ∼0.25′′ (∼35 au). Such
+a companion would need to be on an inclined (nearly polar) orbit
+to misalign the inner disk (Zhu 2019) which would then lead to
+the shadows (Nealon et al. 2019) and variable extinction events
+in the light curves (Ansdell et al. 2020; Sicilia-Aguilar et al.
+2020). It would however not explain the strong localized veloc-
+ity residual feature that we report, which we speculate traces a
+planetary-mass object located at the edge of the dust continuum.
+Detailed modeling of the system is thus needed to assess the
+need for an additional companion. An interesting comparison
+is the CS Cha spectro-binary system (separation ∼7 au) which
+shows similar dust continuum and gas emission at similarly low
+inclination but no departure from Keplerian rotation in the 12CO
+kinematics (Kurtovic et al. 2022).
+6. Conclusions
+In this letter, we present new ALMA observations of the 1.3 mm
+dust continuum and the 12CO (J=2-1) line emission from the
+transition disk around RXJ1604.3–2130 A. The dust continuum
+shows a large cavity enclosing a smaller 12CO cavity. Azimuthal
+Article number, page 5 of 13
+
+A&A proofs: manuscript no. main
+brightness variations in the 12CO line and dust continuum are
+broadly aligned with shadows detected in scattered light (Pinilla
+et al. 2018). Using the discminer package (Izquierdo et al.
+2021), we model the channel-by-channel line emission and cal-
+culate the line-of-sight velocity maps. We report the detection
+of a coherent, localized non-Keplerian feature at R = 41 ± 10 au
+and PA = 280◦ ± 2◦, that is within the continuum ring. While
+broadly aligned with the scattered light shadows, the localized
+non-Keplerian feature cannot be due to changes in temperature.
+Instead, we interpret the kinematical perturbation as tracing the
+presence of a massive companion of Mp ≈ (1.6 − 2.9) Mjup. We
+also detect a tightly wound spiral that extends over 300◦ in az-
+imuth, possibly connected to the localized feature and caused by
+buoyancy resonances driven by planet-disk-interactions. Bend-
+ing of the iso-velocity contours within the gas cavity indicates
+a highly perturbed inner region, possibly related to the pres-
+ence of a misaligned inner disk. However, as the putative planet
+at ∼41 au cannot explain the gas cavity, the low accretion rate
+and the misaligned inner disk, we speculate that another mas-
+sive companion, likely on an inclined orbit, shapes the inner
+∼0.25′′(∼35 au).
+Acknowledgements. We would like to thank the anonymous referee for the
+constructive feedback, as well as Clement Baruteau, Kees Dullemond, Guil-
+laume Laibe and Andrew Winter for helpful discussions. This Letter makes
+use of the following ALMA data: ADS/JAO.ALMA#2017.A.01255.S and
+ADS/JAO.ALMA#2015.1.00964.S. ALMA is a partnership of ESO (represent-
+ing its member states), NSF (USA), and NINS (Japan), together with NRC
+(Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in co-
+operation with the Republic of Chile. The Joint ALMA Observatory is oper-
+ated by ESO, AUI/NRAO, and NAOJ. This project has received funding from
+the European Research Council (ERC) under the European Union’s Horizon
+2020 research and innovation programme (PROTOPLANETS, grant agreement
+No. 101002188). Software: CARTA (Comrie et al. 2021), CASA (McMullin
+et al. 2007), Discminer (Izquierdo et al. 2021), Eddy (Teague 2019a), FilFinder
+(Koch & Rosolowsky 2015), GoFish (Teague 2019b), Matplotlib (Hunter 2007),
+Numpy (van der Walt et al. 2011), Scipy (Virtanen et al. 2020).
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+47
+1 Laboratoire Lagrange, Université Côte d’Azur, CNRS, Observatoire
+de la Côte d’Azur, 06304 Nice, France;
+e-mail: [email protected]
+2 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+3 European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748
+Garching bei München, Germany
+4 Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA
+Leiden, The Netherlands
+5 Universitá degli Studi di Milano, via Celoria 16, 20133 Milano, Italy
+6 Department of Earth, Atmospheric, and Planetary Sciences, Mas-
+sachusetts Institute of Technology, Cambridge, MA 02139, USA
+7 Max Planck Institute for Astronomy, Königstuhl 17, 69117, Heidel-
+berg, Germany
+8 Mullard Space Science Laboratory, University College London,
+Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
+9 Department of Astronomy, University of Florida, Gainesville, FL
+32611, USA
+10 NASA Headquarters, 300 E Street SW, Washington, DC 20546,
+USA
+11 National Radio Astronomy Observatory, Charlottesville, VA 22903,
+USA
+12 The Graduate University for Advanced Studies, SOKENDAI,
+Shonan Village, Hayama, Kanagawa 240-0193, Japan
+13 Departamento de Astronomía, Universidad de Chile, Camino El
+Observatorio 1515, Las Condes, Santiago, Chile
+Article number, page 6 of 13
+
+Stadler et al.: A kinematically-detected planet candidate in a transition disk
+Appendix A: Observations, calibration and imaging
+Table A.1: Summary of the ALMA Band 6 observations of J1604 presented in this paper.
+ID
+EB Code
+Date
+Baselines
+Frequency
+Exp. Time
+PI
+[m]
+[GHz]
+[min]
+2015.1.00964.S
+X412
+2016 Jul 2
+15-704
+217.2-233.4
+8.87
+Oberg
+2017.A.01255.S
+Xb18
+2019 Sep 4
+38-3638
+213.0-230.6
+14.49
+Benisty
+X2fe5
+2021 Apr 29
+15-1263
+14.47
+X4583
+2019 Jul 30
+92-8548
+29.33
+X5f6e
+2019 Jul 31
+92-8548
+29.33
+X104fc
+2021 Sep 27
+70-14362
+29.33
+To self-calibrate our observations, we proceeded as follows. We first flagged the channels containing the line to produce a
+continuum dataset. We centered the individual execution blocks (EBs) by fitting the continuum visibilities with a ring model,
+allowing for a different center and amplitude, enabling us to recover for the phase-shift and amplitude re-scaling to apply to the EBs
+before combining them. To determine a good initial model for the self-calibration, we used multi-scale cleaning with the tclean
+task using a threshold of 2 times the rms noise level of the image. Using the tasks gaincal and applycal, we corrected for phase
+offsets between spectral windows, and between polarizations considering a solution interval of the scan length (solint=inf).
+Executions obtained in 2019 were concatenated and self calibrated together, and similarly for those obtained in 2021. In addition to
+the first found of self calibration, two additional iterations of phase self-calibration were done with solution intervals of 300s and
+180s for the 2019 data, and only one for the 2021 data, with a solution intervals of 360s. For both datasets, a round of amplitude
+self-calibration was applied with solint=inf. The solutions were then applied to the gas data. While these two epochs will be
+analyzed separately for the continuum in a forthcoming paper (Kurtovic et al.), to analyze the gas data, we concatenated them
+after checking that the data do not show significant variation between the two epochs. We imaged the resulting visibilities with
+the tclean task using the multi-scale CLEAN algorithm with scales of 0, 1, 3 and 6 times the beam FWHM, and an elliptic CLEAN
+mask encompassing the disk emission. The 12CO (2-1) molecular line observations are imaged with a robust value of 1.0, a channel
+width of 0.1 km s−1 and masked by 4.0 σ threshold. The data was tapered to 0.′′1 and we used the ’JvM correction’ (Jorsater & van
+Moorsel 1995; Czekala et al. 2021).
+Fig. A.1: Gallery of selected channel maps. Panels show the 12CO data (top row) and best-fit model (middle row) channel maps, together with
+intensity residuals in Kelvin for each channel (bottom row), where in the latter the colorbar has been adjusted such that residuals smaller than
+1σ are white. The beam size is depicted in the lower left corner of each channel. For reference the best-fit systemic velocity was found to be
+vsys = 4.62 km s−1 and the channel spacing is 100 m s−1 .
+Article number, page 7 of 13
+
+_ 4.21 km/s 
+4.41 km/s
+4.61 km/s
+4.81 km/s
+ 5.01 km/s
+ 5.21 km/s
+Data 
+250
+Offset [au]
+0
+-250
+.
+0
+_4.21 km/s
+4.41 km/s
+4.61 km/s
+4.81 km/s
+5.01 km/s
+5.21 km/s
+Model
+250
+[au]
+Offset
+0
+75
+Intensity [K]
+56 
+37
+-250
+18
+:
+4.21 km/s
+4.41 km/s
+5.01km/s
+_5.21 km/s
+4.61km/s
+4.81 km/s
+Residual
+250
+[au]
+Offset
+20
+Residuals [K]
+10
+0
+-250
+-10
+-20
+-250
+250
+Offset [au]A&A proofs: manuscript no. main
+Fig. A.2: Radial profile of the surface brightness for different tracers. Profiles are normalized to the peak of the emission for the 231 GHz
+continuum, CO peak flux, both for the data and discminer model, as well as for the SPHERE scattered light observation. Shaded regions show
+the standard deviation of each azimuthal average. The lines in the lower right corner show the major beam size (resolution) for each profile in the
+corresponding colour.
+Fig. A.3: Azimuthal profiles of the surface brightness, normalized to the peak of the emission. Profiles extracted at an annulus with a width of
+approximately one corresponding beam size centered at 0.56′′ and 0.39′′ for the 231 GHz continuum and the CO peak flux, which both show
+significant azimuthal intensity variations of 34% and 19%, respectively. Shaded regions show the standard deviation of each radial average.
+Fig. A.4: Additional moment maps of the centroid fitting. Panels show the line width ∆V (a), the peak optical depth τ0 (b) and the error of the
+centroid fitting δv0. Note that for τ0 < 1 one can assume the line profile to be well presented by a Gaussian, while for τ0 > 5 the line profile
+has a saturated cored, i.e. a very flat top (see Eq. 2). The beam size is depicted in the lower left corner and only regions where I0 > 5σ with
+σ = 1.1 mJy beam−1 are shown.
+Article number, page 8 of 13
+
+CO peak intensity
+100
+model peak intensity
+Normalised Surface Brightness
+dust continuum
+scattered light
+10-
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+Radius (arcsec)1.2
+12CO at R=(0.39±0.07)"
+Normalised Surface Brightness
+Continuum at R=(0.56±0.02)"
+.0
+0.8
+0.7
+0.6
+-180 -150 -120
+-90
+-60
+-30
+0
+30
+60
+90
+120
+150
+180
+Position Angle (deg)△V (m/s)
+6vo (m/s)
+To
+100
+200
+300
+400
+500
+46810121416
+¥18
+20
+2
+4
+6
+8
+10
+¥12
+14
+16
+18
+20
+0
+2
+2.0 F
+(a)
+(b)
+(c)
+1.5
+1.0
+0.5
+0.0
+Offset (
+-0.5
+-1.0
+-1.5
+D
+-2.0
+2
+0
+0
+2
+0
+2
+2
+Offset (arcsec)
+Offset (arcsec)
+Offset (arcsec)Stadler et al.: A kinematically-detected planet candidate in a transition disk
+Fig. A.5: Polar map of the velocity residuals. Same as Fig. 4, but now overlaid by filamentary structures found by FilFinder. The red and blue
+lines overplotted are the medial axes of the filamentary structures found by the algorithm. To trace the apparent spiral in the residuals, we restricted
+the algorithm to search for filaments in the radial locations r = [0.3, 1.25]′′. For the filamentary detection, we assume a smoothing size of one
+synthesized beam size and a minimum size of 500 pixels for a filament to be considered.
+Fig. A.6: Polar contour map of the centroid residuals. Upper panel shows residuals inside the cavity, middle panel between cavity and outer edge
+of dust ring and lower panel the outer disk. The radial spacing between each contour is ∼1.8 au and the opacity of the lines increase with radius.
+We like to emphasis that the bump between PA≈ −(110 − 70)◦ in the middle panel makes most of the residuals points we detect with the Peak
+Variance method of discminer, which can readily be seen in the left panel of Fig. B.1.
+In Fig. A.7, we show a comparison of velocity residual maps for additional cubes, imaged using different imaging parameters
+to assess the robustness of our detections. We compare residual maps for the same cube as used in the main text, but without JvM-
+correction, and for a cube imaged with a different tapering (0.15" instead of 0.10"). Best-fit Keplerian models were subtracted from
+each of the cubes. As evident from the comparison of the residual maps, the detection of non-Keplerian features reported are robust
+irrespectively of the imaging parameters. However, the detailed morphology of the velocity residual peaks changes with imaging
+Article number, page 9 of 13
+
+Location (0.09i< R ≤ 0.29)"
+Continuum shadows
+400
+Centroid residual (m/s)
+200
+0
+200
+-400
+-270
+-240
+-210
+-180
+-150
+-120
+-90
+09-
+-30
+0
+30
+60
+60
+FLocation (0.29i< R ≤ 0.62)"
+Centroid residual (m/s)
+40
+20
+0
+-20
+-40
+-60
+-270
+-240
+-210
+-180
+-150
+-120
+-90
+-60
+-30
+0
+30
+60
+40 FLocation (0.62 < R ≤ 1.38)"
+Centroid residual (m/s)
+20
+0
+-20
+-40
+-270
+-240
+-210
+-180
+-150
+-120
+-90
+-60
+-30
+0
+30
+60
+Position Angle (deg)0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+270
+major beam size
+60
+dust ring FWHM
+50
+linear spiral
+40
+180
+30
+Postion Angle (degree)
+20
+90
+10
+0
+0
+-20
+-30
+270
+-40
+-50
+60
+180
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+Radius (arcsec)A&A proofs: manuscript no. main
+Fig. A.7: Comparison of the velocity residual maps for different imaging parameters. Left corresponds to the cube used in the main text, while the
+middle panel corresponds to the same cube without JvM-correction. The right panel shows the residual maps for a non-JvM corrected cube with a
+different taper (0.15′′). In all pannels the dust continuum is overlaid in solid contours with equal levels as in Fig. 1. Best-fit Keplerian models were
+subtracted from each of the cubes. The detection of the non-Keplerian features is quite robust irrespectively of the imaging procedure.
+parameters, and as a consequence, the value of the inferred planet location from the discminer analysis. We find that the best-fit
+discminer models to the non-JvM corrected cubes are similar within 3% w.r.t. the best-fit parameters listed in B, with the exception
+of the line slope and some of the peak intensity parameters, that vary up to 15%. While estimating the systematics due to imaging
+parameters is beyond the scope of this letter, Fig. A.7 provides the evidence that the detection of non Keplerian features is robust.
+We measure the rms in a line-free channel to be 2.6 mJy beam−1 and 2.9 mJy beam−1 for the non-JvM corrected cubes with a taper
+of 0.10 and 0.15, respectively.
+Appendix B: Model best fit parameters
+Attribute
+Prescription
+Best-fit parameters
+Centre offset
+xc, yc
+xc = −2.66+0.05
+−0.06 au
+yc = −0.07 ± 0.03 au
+Position angle
+PA
+PA = 258.75+0.06
+−0.05 deg
+-
+Systemic velocity
+vsys
+vsys = 4617.2+0.3
+−0.4 m s−1
+-
+Rotation velocity
+vkep =
+�
+GM⋆
+R
+M⋆ = 1.220 ± 0.001 M⊙
+-
+Ip = Ip0 (R/Rbreak)p0
+R ≤ Rbreak
+Ip0 = 9.388+0.003
+−0.005 mJy pixel−1
+p0 = 1.497+0.004
+−0.005
+Peak intensity
+Ip = Ip0 (R/Rbreak)p1
+Rbreak < R ≤ Rout
+Rbreak = 56.78+0.06
+−0.05 au
+p1 = −0.789 ± 0.001
+Ip = 0
+R < Rout
+Rout = 267.2 ± 0.1 au
+-
+Line width
+Lw = Lw0(R/D0)p
+Lw0 = 0.4097 ± 0.0004 km s−1
+p = −0.592+0.001
+−0.002
+Line slope
+Ls = Ls0(R/D0)p
+Ls0 = 4.569+0.008
+−0.009
+p = −0.454+0.005
+−0.008
+Table B.1: Table of attributes of the discminer model for the 12CO intensity channel maps of the disk around J1604. PA is the
+position angle of the semi-major axis of the disc on the red-shifted side, R the cylindrical radius and D0 = 100 au a normalization
+constant for the line properties. The (down-sampled) pixel size of the model is 8.8 au.
+For the initial emcee run, we use literature values for the position angle and stellar mass (PA=260◦, M⋆ = 1.24 M⊙, Dong et al.
+2017; Manara et al. 2020, respectively). The initial values of the other parameters were found by comparing the overall morphology
+between the data and a prototype model. We performed the MCMC search with 150 walkers which evolved for 2000 steps for an
+initial burn-in stage. We proceeded in two steps. First, we masked the disk region inward of the dust continuum and only fitted
+the outer disk (R > 90 au) to get a robust estimate of the stellar mass and avoid confusion of the code with strongly non-Keplerian
+velocity features in the inner regions. In this run, we interestingly find a strong offset from the disk center in x-direction of −8.0 au. In
+a second step, we fixed the stellar mass and now fitted for the whole disk, masking an inner region corresponding to one major beam
+Article number, page 10 of 13
+
+main cube
+non-JvM
+non-JvM
+taper 0.10
+taper 0.10
+taper 0.15Stadler et al.: A kinematically-detected planet candidate in a transition disk
+size (26 au) in radius where effects of beam smearing are strongest. We run 150 walkers for 20000 steps till we reach convergence,
+resembled by a nearly normal distribution of the walkers. The variance and median of the parameters walkers remain effectively
+unchanged after ∼ 7000 steps. The best-fit parameters are the median of the posterior distributions and given errors are the 16 and
+84 percentiles in the last 5000 steps of the 20000 step run, summarized in Table B.1.
+Fig. B.1: Location of the folded peak velocity residuals. The detected points are shown in azimuth (left) and radius (middle) obtained with the
+Peak Variance method. Colours indicate the 7 different radial clusters specified, where blue peak residual points are within detected significant
+radial cluster. The black crosses are the velocity variances of the clusters plotted at the (R, φ)-location of each cluster center. The centers of the
+accepted clusters (those with peak velocity residuals larger then three times the variance in other clusters) in radius and azimuth are marked with
+black vertical lines in both panels. The right hand plot shows the normal distribution of the peak residual points in a histogram. Note that outliers
+of the distribution are related to the localized perturbation. The maximum value of all peak folded centroid residuals is at 0.39′′ (57 au), its mean
+value is 39 m/s and 1σv = 20 m/s (not to be mistaken with the cluster variances).
+Appendix C: Decomposition and deprojection of velocity components
+To determine the rotation curves of each velocity component, we use the code eddy (Teague 2019a). We follow the method presented
+in Teague et al. (2018b) that uses a Gaussian process to determine the azimuthal vφ and radial vr velocity components along a given
+annulus. To this end, we divide the disk into concentric annuli with a radial width of 1/4 of the synthesized beam (∼ 0.05′′) ranging
+from 0.18′′ to 1.85′′ and extract the velocities over 20 iterations to minimize their standard deviation. To obtain the vertical velocity
+component vz, we use the measured azimuthally averaged profiles of vφ and vr and extend them to produce 2D maps, considering
+the projection of these components along the line of sight:
+vφ, proj = vφ cos (φ) sin (|i|),
+vr, proj = vr sin (φ) sin (i),
+vz, proj = −vz cos (i),
+(C.1)
+where φ is the polar angle in the disk frame (such that φ= 0 corresponds to the red-shifted major axis) and i the inclination of the
+disk. In the case of J1604, the disk rotates clockwise which corresponds to a negative inclination in the above definition. We then
+subtract these maps together with the systemic velocity vsys from the line of sight velocity v0-map (Fig. 2,a) to obtain a map of the
+vertical velocity component vz, proj:
+vz, proj = v0 − vsys − vφ, proj − vr, proj,
+(C.2)
+The radial profile of vz is obtained by deprojecting and azimuthally averaging its 2D velocity map. The radial profiles of the
+deprojected velocity components can be found in Fig. C.1.
+Article number, page 11 of 13
+
+0
+0.40
+acc. cluster centers
+0.10
+.
+80
+80
+2
+0.35
+3
+ Cluster Velocity
+I Residual (m/s)
+70
+70
+(s/w)
+0.08
+5 
+0.30
+6
+Residual
+60
+1g
+60
+0.25
+0.06
+y Variance (
+50
+50
+Folded Centroid
+.
+Centroid
+0.20
+X
+S
+:
+X
+40
+40
+0.15
+0.04
+ (km²/s2)
+Folded (
+C
+.
+30
+X
+30
+0.10
+8
+0.02
+20
+20
+0.05
+X
+X
+:
+X
+LX.
+X
+X
+X
+XK
+0.00
+0.00
+-80
+0.0
+-40
+0
+40
+80
+0.5
+1.0
+1.5
+Azimuth (degree)
+Radius (arcsec)A&A proofs: manuscript no. main
+Fig. C.1: Azimuthally averaged and deprojected azimuthal, radial and vertical velocity components. The radial width of each annulus is 1/4
+synthesized beam size. The error bars are given by the standard deviation for each velocity component averaged over the 20 iterations used.
+Fig. C.2: J1604 deprojected velocity components. It is assumed that all velocities are either azimuthal (left column), radial (central column) or
+vertical (right column). For the azimuthal and radial components wedges along the minor and major axis have been masked as the observations
+are insensitive to these components (see Eq. C.1).In each panel the synthesised beam is shown in the lower left corner.
+Article number, page 12 of 13
+
+6
+dust ring FWHM
+12CO lo peak
+(s/u>y)
+Vkepl
+2
+0.1
+(V - Vkepi)/Vkepl 
+0.0
+-0.1
+0.2
+-0.3
+100
+50
+(s/w)
+0
+-50
+-100
+20
+(s/w)
+0
+-20
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+Radius (arcsec)Vμ (m/s)
+Vr (m/s)
+Vz (m/s)
+-400
+-200
+0
+200
+400
+600
+600
+-400
+-200
+0
+200
+400
+600
+-60
+-40
+-20
+0
+20
+40
+600
+60
+← slower
+faster
+←inwards
+outwards
+← downwards
+upwards→
+2.0
+(b)
+(a)
+(c)
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+-2.0
+2
+1
+0
+-2
+2
+0
+-1
+-2
+2
+1
+0
+-1
+-2
+Offset (arcsec)
+Offset (arcsec)
+Offset (arcsec)Stadler et al.: A kinematically-detected planet candidate in a transition disk
+Appendix D: Derivation of ∆vφ in dependence of azimuthal temperature variations ∆T
+To relate the change in the brightness temperature ∆T of 12CO to variations in the rotational velocity vφ we solve the Navier-Stokes-
+equation in cylindrical coordinates in the φ-direction:
+ρg
+vφ
+R
+∂vφ
+∂φ = 1
+R
+∂p
+∂φ + µ
+� ∂
+∂R
+� 1
+R
+∂
+∂R(Rvφ)
+� 1
+R2
+∂2vφ
+∂φ2
+�
+,
+(D.1)
+with the cylindrical radius R, the gas density ρg and the mean molecular weight µ. In a first step, we assume that the radial variations
+within the chosen annulus are negligible and insert the disk gas pressure in the vertically isothermal assumption p = ρgc2
+s = ρg
+kBT
+µmp ,
+with the Boltzmann constant kB and the proton mass mp. In the second step, we further assume the gas density to be constant along
+the annulus ρg = const. and re-arrange the equation.
+ρg
+vφ
+R
+∂vφ
+∂φ = 1
+R
+∂
+∂φ
+�
+ρg
+kBT
+µmp
+�
++ µ
+R2
+∂2vφ
+∂φ2
+∂vφ
+∂φ −
+µ
+vφRρg
+∂2vφ
+∂φ2 =
+kB
+vφµmp
+∂T
+∂φ
+∂vφ
+∂φ −
+0.4µ
+vφ
+√
+2πΣg
+∂2vφ
+∂φ2 =
+kB
+vφµmp
+∂T
+∂φ
+(D.2)
+In the last step, we inserted the gas midplane density ρg = Σg/(
+√
+2πH) = Σg/(
+√
+2π 0.2R) assuming a disk aspect ratio of H/R = 0.2.
+Assuming that vφ ≈ vkep and Σg ≈ 1g/cm2 (see Fig. 3 of Dong et al. 2017) at the location of the annulus at R ∼ 0.4′′ (58 au), we can
+assess the order of magnitude of the second term on the left hand side of the equation which is only on the order of 10−6. Therefore,
+we neglect the second order derivative and further identify the sound speed cs:
+∆vφ ≈
+kB
+vkep µmp
+T
+T ∆Tφ ≈ c2
+s
+vkep
+∆Tφ
+T
+���� ÷ vkep
+∆vφ
+vkep
+≈
+� cs
+vkep
+�2 ∆Tφ
+T
+≈
+�H
+R
+�2 ∆Tφ
+T
+(D.3)
+The last equation now connects the fractional azimuthal temperature variation ∆Tφ/T to the rotational velocity deviation relative to
+Keplerian ∆vφ/vkep.
+Article number, page 13 of 13
+

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+Pseudo-Goldstone modes and dynamical gap generation from order-by-thermal-disorder
+Subhankar Khatua,1, 2 Michel J. P. Gingras,2 and Jeffrey G. Rau1
+1Department of Physics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4, Canada
+2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
+(Dated: January 31, 2023)
+Accidental ground state degeneracies – those not a consequence of global symmetries of the Hamiltonian
+– are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as “order-by-
+disorder” (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative
+signatures that distinguish ObD from conventional energetic selection. We show that for order-by-thermal-
+disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced
+pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the fer-
+romagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent
+mean-field calculations, we determine the pseudo-Goldstone gap, ∆, and show that at low temperatures it scales
+as the square root of temperature,
+√
+T. We establish that a power-law temperature dependence of the gap is a
+general consequence of ObTD, showing that all key features of this physics can be captured in a simple model
+of a particle moving in an effective potential generated by the fluctuation-induced free energy.
+Strongly competing interactions, or frustration, enhance
+quantum and thermal fluctuations, and undermine the devel-
+opment of conventional magnetic order. The latter can even be
+prevented entirely down to zero temperature, leading to classi-
+cal [1–3] or quantum spin liquids [4–10]. However, additional
+perturbative interactions can relieve the frustration and favor
+the development of long-range order (LRO). Accordingly, the
+majority of spin liquid candidates ultimately evade fate as a
+spin liquid [8, 11]. The ability of these interactions, incon-
+sequential without frustration, to dictate the ground state and
+low-temperature properties of a system is at the root of the
+plethora of exotic phenomena displayed by highly-frustrated
+magnetic materials [10, 12–18].
+This relief of frustration is not always complete. Instead
+of an extensively degenerate manifold, a system can possess
+a sub-extensive accidental ground state degeneracy, unpro-
+tected by symmetry. Classically, this degeneracy can be ro-
+bust to a range of realistic interactions including symmetry-
+allowed two-spin exchange [19]. Here, the role of fluctuations
+is dramatically changed: instead of being detrimental, they
+can lift the classical degeneracy and stabilize order – this is the
+celebrated phenomenon of order-by-disorder (ObD) [20–22].
+While numerous theoretical models have been proposed [20–
+33], there is a paucity of real materials that unambiguously
+harbor ObD [19, 34–37]. The standard strategy for exper-
+imental confirmation is indirect, relying on parametrizing a
+theoretical model of the material, establishing ObD within
+that model, and then validating its predictions for the ordered
+state experimentally.
+While this program has been applied somewhat success-
+fully to a handful of materials [19, 34–37], the inability
+to evince ObD directly, without relying on detailed mod-
+elling, highlights something lacking in our understanding
+of ObD. Clear qualitative, model-independent signatures are
+needed; for example, experimental observation of characteris-
+tic power-laws in heat capacity or transport can diagnose the
+character of low-energy excitations, such as exchange statis-
+tics, dimensionality or their dispersion relations [9, 11, 38,
+39]. Does the presence of ObD exhibit a “smoking-gun” ex-
+perimental signature? This can be difficult or subtle to discern.
+For ObD from quantum fluctuations [21], the formation of an
+ObD spin-wave gap is generally not distinguishable from one
+induced energetically by multi-spin interactions [40–42].
+In this Letter, we identify a clear signature of order-by-
+thermal-disorder (ObTD): a dynamically generated gap grow-
+ing as the square root of temperature.
+We investigate this
+gapped “pseudo-Goldstone” (PG) mode [44–46] in a minimal
+2D classical spin model exhibiting ObTD, the ferromagnetic
+Heisenberg-compass model on a square lattice, belonging to
+a class of models relevant to Mott insulators with strong spin-
+orbit coupling [47–55]. Through spin-dynamics simulations,
+we determine the PG gap, ∆, and show it varies with tempera-
+ture as ∆ ∝
+√
+T, in quantitative agreement with self-consistent
+mean-field theory (SCMFT). This mode is well-defined, with
+the linewidth, Γ, due to thermal broadening, Γ ∝ T 2 ≪ ∆. We
+further demonstrate that our key results can be captured by an
+effective description of a particle moving in a potential gener-
+ated by the ���uctuation-induced free energy. Using this picture,
+we argue that the temperature dependence of the PG gap,
+√
+T
+(T) for type-I (II) PG modes [56], is universal, applicable to
+any system exhibiting ObTD. Finally, due to the low dimen-
+sionality [57], ObTD faces a subtle competition against poten-
+tially infrared-divergent fluctuations [58, 59]. While ObTD
+ultimately prevails, and true LRO develops, the magnetization
+displays logarithmic corrections at low temperature, a rem-
+nant of the diverging infrared fluctuations.
+Model.—
+We
+consider
+the
+classical
+ferromagnetic
+Heisenberg-compass model on a square lattice
+H =
+�
+r
+�
+−J
+�
+δ=ˆx,ˆy
+Sr · Sr+δ − K
+�
+S x
+rS x
+r+ˆx + S y
+rS y
+r+ˆy
+��
+,
+(1)
+where Sr ≡ (S x
+r, S y
+r, S z
+r) is a unit vector at site r, and δ =
+ˆx, ˆy denote the nearest-neighbor bond directions. We consider
+ferromagnetic Heisenberg and compass interactions with J >
+0, K > 0 (see SM [60] for a discussion of other signs) and
+with J the unit of energy, setting J ≡ ℏ ≡ kB ≡ 1 throughout.
+For K = 0, the model [Eq.(1)] is the well-known Heisen-
+berg ferromagnet with uniform ferromagnetic ground states
+of arbitrary direction, Sr = ˆn, related by global spin-rotation
+arXiv:2301.11948v1  [cond-mat.str-el]  27 Jan 2023
+
+2
+−π/4
+0
+π/4
+φ
+0.0
+0.2
+0.4
+0.6
+0.8
+P(φ)
+(a)
+L = 14
+L = 10
+L = 6
+[00]
+[π0]
+[ππ]
+[00]
+[0π]
+0
+5
+10
+15
+20
+25
+ω
+(b)
+[00]
+[π0]
+[ππ]
+[0π]
+kx
+ky
+0
+2
+4
+0
+200
+0
+1
+2
+3
+4
+5
+0.0
+0.2
+0.4
+0.6
+ω
+S(0,ω) [arb.]
+(c)
+T = 0.040
+T = 0.032
+T = 0.024
+T = 0.016
+T = 0.008
+[00]
+[π0]
+[ππ]
+[00]
+[0π]
+0
+5
+10
+15
+20
+25
+ω
+(d)
+0
+5
+Spin-dynamics
+LSWT
+SCMFT
+FIG. 1. (a) Probability distribution, P(φ), of the angle, φ, characterizing the direction of the net magnetization obtained using MC simulations
+with K = 5 at T = 0.4 for several system sizes, L. Due to C4 symmetry, P(φ) is shown for φ ∈ [−π/4, π/4]. (b) Dynamical structure factor,
+S(k, ω) obtained from spin-dynamics simulations for L = 100 with K = 5 at T = 0.4 along a path through the Brillouin zone (see left inset).
+Overall intensity is arbitrary. (Right inset) Spectrum near [00] showing the PG gap [43]. (c) Dynamical structure factor at k = 0, S(0, ω),
+obtained from spin-dynamics simulations for L = 40 at various temperatures with K = 5. Overall intensity is arbitrary. (d) Excitation spectrum
+along the same path as in panel-(b) from the LSWT, SCMFT, and spin-dynamics simulations with K = 5 for L = 100 at T = 0.4. The
+spin-dynamics spectrum tracks the frequencies of maximum of S(k, ω). The inset highlights a small region near [00], showing the PG mode.
+symmetry. For K > 0, this symmetry is absent and H in
+Eq. (1) is minimized by any uniform magnetization in the
+ˆx − ˆy plane. These ground states are characterized by an an-
+gle φ ∈ [0, 2π) with Sr = cos φ ˆx + sin φ ˆy. Unlike the pure
+Heisenberg ferromagnet, these are only accidentally degener-
+ate, as the continuous in-plane spin rotations connecting them
+do not preserve the anisotropic compass term. However, a dis-
+crete C4 symmetry about the ˆz axis and C2 symmetries about
+the ˆx and ˆy axes still remain.
+Simulations.— We first show that this model exhibits ObTD
+via Monte Carlo (MC) simulations on a lattice with N = L2
+sites. To expose the state selection, we construct a proba-
+bility distribution for magnetization direction, encoded in φ,
+P(φ), using a sample of thermalized states (see SM [60]). As
+shown in Fig. 1(a), P(φ) exhibits maxima at φ = 0, π/2, π,
+3π/2, corresponding to ferromagnetic ground states with ˆn
+along the ±ˆx, ±ˆy directions. At low temperatures, fluctua-
+tions thus select four discrete ground states via ObTD from a
+one-parameter manifold of states.
+We now consider the classical dynamics to examine the as-
+sociated PG mode. The equation of motion for the classical
+spins is the Landau-Lifshitz equation [61], dSi/dt = Br × Sr,
+describing precession about the exchange field, Br, produced
+by neighboring spins
+Br ≡ −
+�
+δ=±ˆx,±ˆy
+�
+JSr+δ + KS δ
+r+δδ
+�
+.
+(2)
+Starting with states drawn via MC sampling at temperature T,
+we numerically integrate the Landau-Lifshitz equations, and
+compute the dynamical structure factor, S(k, ω) = ⟨|Sk(ω)|2⟩,
+where Sk(ω) is the Fourier transform of spins, and ⟨· · ·⟩ de-
+notes averaging over the initial states [60]. Results for S(k, ω)
+at a representative T and K [60] are shown in Fig. 1(b),
+exhibiting sharp spin-waves with a nearly gapless mode at
+k = 0. Closer examination reveals a well-defined gap, as
+highlighted in the top right inset of Fig. 1(b) – this is the PG
+gap.
+To determine the PG gap quantitatively, we consider a cut
+of the structure factor at k = 0, i.e., S(0, ω). As the PG
+gap is much smaller than the bandwidth of the spectrum [see
+Fig. 1(b)], a significantly higher frequency resolution is re-
+quired to accurately compute the gap [60], so a much longer
+integration time window is necessary. Cuts, S(0, ω), for sev-
+eral temperatures are presented in Fig. 1(c), with the peak lo-
+
+3
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+T
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+∆
+0.00
+0.25
+0.50
+0.75
+1.00
+T
+0.0
+0.2
+0.4
+Γ
+Spin-dynamics
+SCMFT
+SCMFT asymptotic limit
+FIG. 2. Pseudo-Goldstone gap, ∆, as a function of temperature from
+spin-dynamics simulations with K = 5. The data is well-described
+by the fit ∆ = 2.46242
+√
+T − 3.21907 T 3/2. The SCMFT gap agrees
+with it quantitatively and provides the asymptotic T → 0 scaling,
+2.46147
+√
+T. (Inset) Linewidth of the PG mode, Γ, as a function of
+temperature from spin-dynamics simulations. It is well described by
+the fit, Γ = 0.709286 T 2 − 0.329751 T 3. All data have been extrapo-
+lated in the system size to the thermodynamic limit [60].
+cation indicating the PG gap (see SM [60]). The temperature
+dependence of ∆ is shown in Fig. 2. The leading contribution
+to the PG gap scales as the square root of temperature, van-
+ishing as T → 0, and is well-described by the fit ∆ ∼ 2.46
+√
+T.
+The thermal broadening of the spectrum induces a finite
+width to all excitations, including the PG mode. The PG mode
+linewidth, Γ, can be obtained from the full-width at half max-
+imum of S(0, ω) [see Fig. 1(c)] as a function of temperature.
+The inset in Fig. 2 shows that Γ ∝ T 2 at low temperatures
+(see SM [60]). Since Γ ≪ ∆ as T → 0, this PG mode is
+well-defined.
+Spin-wave analysis.— The simulations have revealed that
+the system has LRO and hosts a PG excitation, where the
+PG gap and linewidth scale with temperature as
+√
+T and
+T 2, respectively.
+To understand how these scaling laws
+arise, we consider a spin-wave analysis about the ordered
+state [62]. Since tackling spin-wave interactions is difficult
+within a purely classical approach [63–65], we follow the
+more widely used and computationally convenient quantum
+spin-wave analysis [66–68], taking the classical limit only at
+the end.
+We first discuss the spectrum and state selection due to
+ObTD in linear spin-wave theory (LSWT). Expanding about a
+classical ground state (parametrized by φ) using the Holstein-
+Primakoff (HP) transformation [62], we obtain to O(S )
+H2 =
+�
+k
+�
+Aka†
+kak + 1
+2!
+�
+Bka†
+ka†
+−k + H.c.
+��
+,
+(3)
+where ak denotes the bosonic annihilation operator at wave
+vector k, and Ak and Bk depend on φ, J, and K (see SM [60]).
+H2 in Eq. (3) can be diagonalized by a Bogoliubov transfor-
+mation [62], giving spin-wave energies ωk =
+�
+A2
+k − B2
+k. As
+the spectrum depends on the ground state angle φ, fluctuations
+can lift the accidental classical degeneracy. To examine state
+selection due to ObTD, we search for the ground states where
+the free energy is minimal. Starting with the quantum free en-
+ergy Fqu = 1
+2
+�
+k ωk + T �
+k ln
+�
+1 − e−ωk/T�
+, the classical limit
+T ≫ ωk yields F = T �
+k ln ωk [69]. This classical free en-
+ergy has four minima at φ = 0, π/2, π, 3π/2 – establishing
+selection by ObTD, in agreement with the MC results.
+Within LSWT, quantum and classical calculations give the
+same spectrum, ωk [22]. This spectrum, calculated about φ =
+0, exhibits a gapless mode at k = 0 as shown in Fig. 1(d). To
+obtain a PG gap, spin-wave interactions must be included, as
+we next discuss.
+Interacting spin waves.— Performing the HP expansion to
+next order in 1/S , the LSWT Hamiltonian [Eq. (3)] is aug-
+mented by interaction terms. Three-boson interactions are ab-
+sent due to a C2 symmetry about the ordering direction, leav-
+ing only terms quartic in the bosons at O(S 0) (see SM [60]).
+To treat this interacting problem, we adopt a mean-field ap-
+proach [66, 67], decoupling the quartic terms into products
+of quadratic terms and thermal averages of two-boson oper-
+ators. Following this procedure, the new effective quadratic
+Hamiltonian mirrors Eq. (3), but with Ak and Bk replaced with
+(Ak + δAk) and (Bk + δBk). These corrections are
+δAk = 1
+N
+�
+q
+�
+Vk,q,0⟨a†
+qaq⟩ + 1
+2
+�
+Dq,−q,k⟨a†
+qa†
+−q⟩ + c.c.
+��
+,
+δBk = 1
+N
+�
+q
+�
+Dk,−k,q⟨a†
+qaq⟩ + 1
+2Vq,−q,k−q⟨aqa−q⟩
+�
+,
+(4)
+where Vk1,k2,k3 and Dk1,k2,k3 are the coefficients for the 2-2
+and 3-1 magnon scattering terms at O(S 0) [60], and ⟨· · ·⟩ is
+a thermal average. When these averages are computed using
+LSWT [Eq. (3)], the corrections [Eq. (4)] reproduce leading
+order perturbation theory [70, 71]. However, because of the
+gapless mode, these individual δAk and δBk diverge in the
+classical limit and perturbation theory breaks down [60].
+To resolve these divergences, we perform the averages in
+Eq. (4) using SCMFT, obtaining a renormalized spectrum, Ωk
+(see SM [60]). Explicitly, ⟨a†
+qaq⟩ and ⟨a†
+qa†
+−q⟩ are, classically,
+computed self-consistently (until convergence) using Eq. (4)
+and
+⟨a†
+kak⟩ = T(Ak + δAk)
+Ω2
+k
+,
+⟨aka−k�� = −T(Bk + δBk)
+Ω2
+k
+,
+(5)
+where Ωk =
+�
+(Ak + δAk)2 − (Bk + δBk)2 and ⟨aka−k⟩ =
+⟨a†
+ka†
+−k⟩.
+The SCMFT spectrum Ωk, plotted in Fig. 1(d), exhibits a
+clear gap at k = 0. The PG mode, gapless in LSWT, has now
+become gapped due to magnon-magnon interactions. Excel-
+lent agreement between the spectra from SCMFT and spin-
+dynamics simulations is observed across the full Brillouin
+zone [see Fig. 1(d)]. The temperature dependences of ∆ from
+the two approaches in Fig. 2 agree quantitatively, with identi-
+cal
+√
+T scaling as T → 0. This is a key result of this work,
+establishing a clear spectral signature of ObTD.
+
+4
+While the SCMFT is successful in describing the excita-
+tion energies, it does not address thermal broadening, since
+δAk and δBk are real, giving an infinite magnon lifetime.
+To obtain a finite linewidth, perturbation theory must be car-
+ried out to higher order. We expect that δA0 ≡ δAk=0 and
+δB0 ≡ δBk=0, interpreted as contributions to the magnon self-
+energy [60], can be expanded in T as δA0 = a1T + a2T 2 + · · ·
+and δB0 = b1T + b2T 2 + · · · . Since |A0| = |B0|, reflecting
+the gapless LSWT spectrum, and a1, b1 [the O(T) corrections
+in Eq. (4)] are real; any imaginary part, and thus finite life-
+time, must arise from a2 or b2. Expanding Ω0 ≡ Ωk=0 in T
+yields Im Ω0 ≈ (Im a2) T 2 + · · · (see SM [60]). The real part,
+Re Ω0, maintains its leading
+√
+T dependence (providing the
+PG gap) while Im Ω0, giving the linewidth, has a leading T 2
+dependence, consistent with the simulation results (see inset
+of Fig. 2).
+Effective description.— We now present an effective de-
+scription capturing the key aspects of the PG mode in a
+significantly simpler language and with broader applicabil-
+ity, adapting an approach formulated for order-by-quantum-
+disorder (ObQD) [72].
+We consider small uniform devi-
+ations from a classical ground state (say φ
+=
+0) with
+Sr
+≈ (
+�
+1 − φ2 − θ2, φ, θ), accurate to quadratic order in φ
+and θ, where φ is the soft mode and θ its conjugate momentum.
+For small φ and θ, φ ≈ 1
+N
+�
+r S y
+r and θ ≈ 1
+N
+�
+r S z
+r, with Pois-
+son bracket {φ, θ} = 1/N. For this configuration, we define an
+effective free energy Feff(θ, φ) = Ecl(θ) − TS (φ), where Ecl(θ)
+is the classical cost of nonzero θ and S (φ) = − �
+k ln ωk(φ)
+is the entropy. For small θ and φ, Feff can be expanded as
+Feff ≈ 1
+2N
+�
+Cθθ2 + Cφφ2�
+, where Cθ = (∂2Feff/∂θ2)/N = 2K
+and Cφ = (∂2Feff/∂φ2)/N. Taking Feff as an effective Hamil-
+tonian, the equations of motion [73] for θ and φ are
+∂φ
+∂t = + 1
+N
+∂Feff
+∂θ
+= +Cθθ,
+∂θ
+∂t = − 1
+N
+∂Feff
+∂φ
+= −Cφφ,
+(6)
+describing a harmonic oscillator. We identify the PG gap as
+its frequency, ∆ = �CθCφ. Remarkably, the
+√
+T dependence
+of the PG gap is recovered, since Cφ is O(T) and Cθ is O(1).
+The curvature Cφ can be calculated within LSWT, yielding a
+frequency 2.46147
+√
+T for K = 5 – exactly the PG gap found
+in SCMFT as T → 0 and in agreement with the spin-dynamics
+simulations (see Fig. 2).
+While formulated for the Heisenberg-compass model, this
+line of argument can be deployed to obtain the PG gap
+for any spin model exhibiting ObTD. A proof of this state-
+ment, following the strategy of Ref. [72], will be reported
+elsewhere [74].
+For type-I PG modes (ω ∝ |k|, as in
+the Heisenberg-compass model) ∆ ∝
+√
+T, while for type-II
+modes (ω ∝ |k|2), both Cφ, Cφ are O(T) and thus ∆ ∝ T.
+Consequences of MWH divergence.— The ability to obtain
+the PG gap from LSWT presents a puzzle: the perturbative
+corrections δA0 and δB0 diverge logarithmically with system
+size [57], just as in the MWH theorem [58, 59]. How then
+do the curvatures of Feff avoid these singularities and give the
+correct scaling? An analysis of the infrared divergences [60]
+shows that while δA0 and δB0 are singular, δA0 + δB0, which
+determines the leading contribution to the PG gap, is finite,
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+T
+−0.16
+−0.15
+−0.14
+−0.13
+−0.12
+∂M/∂T
+0.0
+0.1
+0.2
+0.3
+T
+0.96
+0.98
+1.00
+M
+SCMFT
+Monte Carlo
+FIG. 3. Derivative of magnetization with respect to temperature,
+∂M/∂T, as a function of temperature for L = 60, K = 5 using MC
+simulation and SCMFT. MC data is well-described by a fit motivated
+by SCMFT [60], −0.09815 − 0.03563T + 0.01485 ln T. A similar
+fit to SCMFT data yields −0.09631 − 0.01494T + 0.01491 ln T. The
+inset shows M as a function of temperature for the same parameters.
+MC error bars on M are smaller than the symbol size.
+and reproduces the result from Eq. (6). However, divergences
+in higher order terms do not cancel, and must be cured self-
+consistently [60].
+While these divergences are mostly benign for the PG gap,
+they appear more dramatically in other quantities, like the
+magnetization, M = 1 − 1
+N
+�
+k⟨a†
+kak⟩. Here, the thermal popu-
+lation, ⟨a†
+kak⟩ diverges in LSWT, rendering SCMFT necessary
+to obtain meaningful results. In SCMFT, the PG gap provides
+an infrared cutoff ℓ ∼ 1/∆ ∝ 1/
+√
+T, giving a logarithmic con-
+tribution to M scaling as ∝ Tln T as T → 0 [60]. The presence
+of this term can be diagnosed from ∂M/∂T, which exhibits a
+logarithmic singularity as T → 0 for both the MC simulations
+and SCMFT (see Fig. 3).
+Outlook.— Our analysis of the PG gap will provide a deeper
+understanding of real materials exhibiting ObD. The existence
+of PG modes has been used to diagnose ObD, for example
+in the compounds Fe2Ca3(GeO4)3 [34], Sr2Cu3O4Cl2 [35]
+and Er2Ti2O7 [36, 41, 75].
+In such materials, the ObQD
+gap likely dominates the ObTD-induced gap discussed in this
+work. However, in systems where the effect of ObQD is weak
+or the degrees of freedom are sufficiently classical, ObTD
+can resurface as the leading selection effect. For example,
+our results may shed light on the rapidly growing family of
+two-dimensional van der Waals (vdW) ferromagnets [76–78]
+where the ObQD gap is expected to be small and thus the gap
+induced by thermal fluctuations may be more significant. Ad-
+ditionally, while reaching the classical thermal regime is chal-
+lenging in magnetic materials (due to small spin length S ),
+it may be more accessible in other platforms such as those
+involving lattice vibrations [79, 80], dipole-coupled nanocon-
+fined molecular rotors [81–84] or artificial mesoscale mag-
+netic crystals [85–88]. Whether ObTD can be realized in such
+topical systems, and how to detect the temperature dependent
+PG gap, are open questions; our approach provides a theoret-
+
+5
+ical framework and guidance for future experimental studies
+in this promising area of research.
+ACKNOWLEDGMENTS
+We thank Itamar Aharony, Kristian Tyn Kai Chung, Alex
+Hickey, Daniel Lozano-Gómez, and Darren Pereira for use-
+ful discussions. We acknowledge the use of computational
+resources provided by Digital Research Alliance of Canada.
+This research was funded by the NSERC of Canada (MJPG,
+JGR) and the Canada Research Chair Program (MJPG, Tier
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+Supplemental Material for “Pseudo-Goldstone modes and dynamical gap generation from
+order-by-thermal-disorder”
+Subhankar Khatua,1, 2 Michel J. P. Gingras,2 and Jeffrey G. Rau1
+1Department of Physics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4, Canada
+2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
+(Dated: January 27, 2023)
+I.
+DETAILS OF MONTE CARLO SIMULATIONS
+A.
+Details of Monte Carlo procedure
+The Monte Carlo (MC) simulations described in the main
+text are based on adaptive single-site Metropolis moves [1],
+combined with over-relaxation moves [2]. A typical single-
+site Metropolis move involves randomly selecting a spin and
+changing its orientation to a random direction.
+However,
+at low temperature, most such moves result in configura-
+tions that are of much higher energies and thus rejected [3].
+Therefore, we follow an adaptive approach that selects a
+spin randomly and changes its orientation to a Gaussian
+distributed random direction within a solid-angle of certain
+width. The solid-angle-width changes adaptively to ensure
+that the update-acceptance rate remains close to 50% at each
+temperature (see Ref. [1] for details).
+The over-relaxation
+move rotates a randomly selected spin by a random angle
+about its local exchange field. This move is energy-conserving
+and thus always accepted. We define a Monte Carlo sweep
+at a certain temperature as a combination of N (total num-
+ber of spins) random successive adaptive single-site Metropo-
+lis moves with each followed by five random over-relaxation
+moves. All the simulation results discussed in the main text
+have been obtained by considering periodic boundary condi-
+tions on the square lattice of size L by L and N = L2 spins. As
+in the main text, we use units such that J = ℏ = kB = 1.
+B.
+Simulation details of the order parameter distribution
+Starting from a random spin configuration at high tempera-
+ture, T = 10 (larger than K) where the system is in the param-
+agnetic phase, we slowly cool down in steps of size δT = 0.1
+to a final temperature T = 0.4 (much smaller than K). At
+each temperature, we perform 105 MC sweeps to equilibrate
+the system. Finally, at T = 0.4, after equilibration, we record
+the net magnetization-per-spin over 106 MC samples, leaving
+five MC sweeps in between two consecutive measurements.
+From the net magnetization per spin, M = (Mx, My, Mz), we
+calculate ϕ = arctan(My/Mx), computing a distribution for ϕ.
+Since the ferromagnetic Heisenberg-compass model has a C4
+rotation symmetry in the ˆx− ˆy plane, we symmetrize the distri-
+bution by shifting the data by π/2, π, and 3π/2 , i.e., add π/2,
+π, and 3π/2 to each entry of the dataset. We have plotted the
+final dataset as a probability density, P(ϕ) for ϕ ∈ [−π/4, π/4]
+with 50 bins for three different system sizes, N = 62, 102,
+and 142 in Fig. 1(a) in the main text. We have chosen a large
+value for K, i.e., K = 5, for all simulations in order to obtain
+a strong selection effect at accessible system sizes. For the
+gross spectral features, the largest system size considered for
+spin-dynamics simulations was N = 1002, while for detailed
+features, such as the temperature dependence of the pseudo-
+Goldstone (PG) gap, up to N = 402 was used. Had smaller
+values of K been used, all the MC simulations, as well as
+spin-dynamics simulations, would have had to be performed
+for much larger system sizes to obtain results that converge
+when system size is extrapolated to the thermodynamic limit
+(N → ∞).
+C.
+Simulation details of magnetization and its derivative with
+respect to temperature
+Independently at each temperature T, 5 × 105 MC sweeps
+are performed on a perfectly aligned ferromagnetic spin con-
+figuration along ˆx for equilibration, followed by 3 × 106 suc-
+cessive MC sweeps to measure the net magnetization along
+ˆx (M), energy (E), and their product (EM). Their product is
+recorded in order to calculate the derivative of the magnetiza-
+tion with respect to temperature, given by
+∂M
+∂T ≡ ⟨EM⟩ − ⟨E⟩⟨M⟩
+T 2
+,
+(S1)
+where ⟨x⟩ is the MC thermal average of quantity x. To es-
+timate the statistical errors on static quantities, the 3 × 106
+measurements are divided into 30 blocks, and then resampled
+using the standard bootstrap method [3]. Typically, O(103)
+bootstrap samples were generated from these blocks to esti-
+mate the statistical errors. In Fig. 3 of the main text, the error
+bars shown correspond to twice the standard deviation esti-
+mated via bootstrap.
+II.
+DETAILS OF SPIN-DYNAMICS SIMULATIONS
+Numerical integrations of the Landau-Lifshitz equations
+have been done using an adaptive step size RK5(4) Dormand-
+Prince integrator [4] from the Boost-Odeint C++ library [5,
+6]. The initial spin configurations for the numerical integra-
+tion are generated from MC simulations described in Sec. I.
+To obtain the results shown in Fig. 1(b) in the main text,
+we perform 5 × 105 equilibration MC sweeps on a perfectly
+aligned ferromagnetic configuration along ˆx at T = 0.4. Start-
+ing from the final state, we perform another 15 × 103 MC
+sweeps for 350 independent parallel runs to generate well-
+equilibrated configurations at T = 0.4. Next, we feed each
+
+9
+2
+0.000
+0.001
+0.002
+0.003
+0.004
+1/L2
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+∆
+T = 0.040
+T = 0.032
+T = 0.024
+T = 0.016
+T = 0.008
+FIG. S1.
+Finite size scaling of the PG gap obtained from spin-
+dynamics simulations for several temperatures (K = 5).
+0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
+1/L2
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+∆
+T = 0.040
+T = 0.032
+T = 0.024
+T = 0.016
+T = 0.008
+FIG. S2. Finite size scaling of the PG gap obtained using SCMFT
+for several temperatures (K = 5).
+of these 350 configurations into the Dormand-Prince integra-
+tor as an initial state and integrate to a final time, tmax = 50.
+The error tolerance of the integrator is set to 10−8, such that
+the energy-per-spin and individual spin lengths are conserved
+to at least one part in 107 and 1010, respectively. In each of
+these independent 350 integrations, we calculate the Fourier
+transform of the spin configurations in space and time, S(k, ω)
+using FFTW++ [7] and then compute the dynamical structure
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+1/L
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Γ
+T = 1.000
+T = 0.800
+T = 0.600
+T = 0.400
+T = 0.200
+T = 0.016
+FIG. S3. Finite size scaling of the PG linewidth obtained from spin-
+dynamics simulations for several temperatures (K = 5).
+factor, S(k, ω) = |S(k, ω)|2, finally taking an average of the
+structure factors found from the 350 initial configurations to
+obtain Fig. 1(b).
+The results in Fig. 1(c) in the main text are obtained
+as follows: The system is initialized in a perfectly aligned
+ferromagnetic configuration along ˆx at T
+= 0.0004, and
+slowly warmed up in steps of δT = 0.0004 to a temper-
+ature T
+= 0.04.
+At each temperature, we perform 105
+equilibration MC sweeps, generating a configuration at T =
+0.008, 0.016, 0.024, 0.032, 0.04.
+At each of these tempera-
+tures, we then apply the following procedure. Starting from
+a given spin configuration, say at T = 0.008, we generate
+a total of 2 × 103 configurations independently by perform-
+ing 105 MC sweeps. Each of these configurations is fed into
+the Dormand-Prince integrator independently to integrate to
+a final time, tmax = 2500. Note that tmax here is taken to be
+much larger than the tmax = 50 value used to obtain the re-
+sults shown in Fig. 1(b). As discussed in the main text, to
+determine the PG gap, ∆, and linewidth, Γ, a much higher
+frequency resolution is needed and thus the total integration
+time must be larger. The error tolerance of the integrator is
+set to 10−10, such that the energy-per-spin and spin-length are
+conserved to at least one part in 108 and 1010, respectively.
+After the time evolution, we compute the Fourier transform of
+the spin configurations in space and time using FFTW++ and
+then compute the dynamical structure factor, S(k, ω). Finally,
+we perform an average over the 2 × 103 initial spin config-
+urations to obtain the average dynamical structure factor at
+T = 0.008. In Fig. 1(c), we show only a cut of the average dy-
+namical structure factor at the zone center, S(0, ω). To clearly
+visualize S(0, ω) at several different temperatures in a single
+plot, we stagger them on the y-axis with a constant spacing
+between the S(0, ω) data at two consecutive temperatures.
+To obtain the results shown in Fig. 2 of the main text,
+we proceed as follows: At each temperature, we follow the
+same method as described for Fig. 1(c) in the previous para-
+graph and compute S(0, ω) for several system sizes, L =
+20, 24, 28, 32, 36, and 40. To find the gap and linewidth for
+each system size, we fit each data to a Gaussian (a Gaussian
+lineshape fits the data in the range T ≤ 0.04 best, compared to,
+e.g., a Lorentzian). The center of the Gaussian is used to de-
+fine the PG gap and the full-width at half maximum (FWHM)
+of the Gaussian, i.e., 2.355σ (standard deviation), is taken as
+the PG linewidth. Then, finite size L-dependent PG gaps and
+linewidths are then extrapolated in system size (L → ∞) to
+obtain the corresponding values in the thermodynamic limit.
+Finite size scaling of the PG gap is shown in Fig. S1. The finite
+size scaling of the PG gap obtained using the self-consistent
+mean-field theory (SCMFT) is shown in Fig. S2 (See Sec. III E
+for details).
+At very low temperatures, e.g., T ≤ 0.04, where S(0, ω)
+falls very sharply away from the center of the peak, a Gaus-
+sian lineshape is a natural choice. However, as temperature
+increases further, S(0, ω) shows more pronounced tails and a
+Lorentzian lineshape was found to provide a better descrip-
+tion of the data. Finite size scaling of the PG linewidth is
+shown in Fig. S3. At T = 0.016, the PG linewidths for differ-
+ent system sizes are found by fitting to a Gaussian while for
+
+10
+3
+the remaining temperatures in Fig. S3, the PG linewidths are
+found from fitting to Lorentzian (via the FWHM of the corre-
+sponding Lorentzian). Finite size scaling reveals that at very
+low temperatures, the PG linewidth scales almost linearly with
+1/L with the scaling becoming quadratic in 1/L as tempera-
+ture increases (see Fig. S3).
+III.
+SPIN WAVE THEORY
+Here, we elaborate on the formalism for interacting spin
+waves in the ferromagnetic Heisenberg-compass model on the
+square lattice. We consider the Heisenberg-compass model
+Hamiltonian
+H = −
+�
+rδ
+�
+JSr · Sr+δ + KS δ
+rS δ
+r+δ
+�
+≡
+�
+rδ
+S⊺
+r JδSr+δ,
+(S2)
+where δ = ˆx, ˆy denotes the nearest-neighbour (horizontal and
+vertical) bond directions. For J > 0 and K > 0, the classical
+ground state is ferromagnetic and has an accidental degener-
+acy parametrized by an angle ϕ
+Sr = S (cos ϕ ˆx + sin ϕ ˆy).
+(S3)
+For small |K| and K < 0, one finds only a (symmetry-
+enforced) discrete degeneracy, with Sr = ±S ˆz. For large |K|
+and K < 0, the ground state is described by an XY-stripe phase
+parametrized by a single angle whose two extreme limits are
+X-stripe phase (i.e., all spins are lying along the ˆx axis, ar-
+ranging themselves antiferromagnetically along the ˆx axis and
+ferromagnetically along the ˆy axis) and Y-stripe phase (i.e., all
+spins are lying along the ˆy axis, arranging themselves antifer-
+romagnetically along the ˆy axis and ferromagnetically along
+the ˆx axis). The phases for J < 0 can be obtained by map-
+ping Sr → (−1)rSr which alternates on the two sublattices.
+We note that the dynamics however differ between J > 0 and
+J < 0, since the sign change on one sublattice is not a canoni-
+cal transformation.
+Returning to the J > 0, K > 0 case, we define a frame
+aligned with the ground state with angle ϕ
+ˆex = − sin ϕ ˆx + cos ϕ ˆy,
+ˆey = ˆz,
+ˆez = cos ϕ ˆx + sin ϕ ˆy,
+as well as ˆe± ≡ (ˆex ± iˆey)/
+√
+2 and ˆe0 ≡
+ˆez.
+We then
+have the local exchanges Jµν
+δ
+= ˆe⊺
+µ Jδˆeν. The Fourier trans-
+forms of the exchange matrices, Jµν
+δ , are defined as J k ≡
+�
+δ 2 cos (k · δ)J δ where the fact that −δ and δ are equivalent
+has been used. Explicitly, these are given by
+J+−
+k
+= −
+�
+2J + Ksin2ϕ
+�
+cos kx −
+�
+2J + Kcos2ϕ
+�
+cos ky,
+J00
+k = −2
+�
+J + Kcos2ϕ
+�
+cos kx − 2
+�
+J + Ksin2ϕ
+�
+cos ky,
+J++
+k
+= −
+�
+Ksin2ϕ
+�
+cos kx −
+�
+Kcos2ϕ
+�
+cos ky,
+J0±
+k
+= − K
+√
+2
+sin (2ϕ)
+�
+cos ky − cos kx
+�
+,
+with J−+
+k
+= [J+−
+k ]∗, J−−
+k
+= [J++
+k ]∗ and J0±
+k
+= J±0
+k . Note
+that J00
+0
+= −2(2J + K). For one of the four ground states
+selected by order-by-thermal-disorder (ObTD), e.g. ϕ = 0,
+these Jµν
+k are given by
+J+−
+k
+= −2J cos kx − (2J + K) cos ky,
+J00
+k = −2 (J + K) cos kx − 2J cos ky,
+J++
+k
+= −K cos ky,
+where J0±
+k
+= 0. Performing the usual Holstein-Primakoff
+expansion [8] to O(S 0) on this model yields [9]
+H ≈ E0 + H2 + �H4,2−2 + H4,3−1 + H4,1−3
+� + · · · ,
+(S4)
+where we have defined the constant classical part E0
+=
+−NS 2(2J + K) [at O(S 2)] and
+H2 =
+�
+k
+�
+Aka†
+kak + 1
+2!
+�
+Bka†
+ka†
+−k + B∗
+ka−kak
+��
+,
+(S5a)
+H4,2−2 = 1
+N
+�
+kk′q
+1
+(2!)2 Vk,k′,qa†
+k+qa†
+k′−qak′ak,
+(S5b)
+H4,3−1 = 1
+N
+�
+kk′q
+1
+3!Dk,k′,qa†
+ka†
+k′a†
+qak+k′+q = H†
+4,1−3.
+(S5c)
+This incldues the quadratic parts [at O(S )] in H2 as well as
+the quartic parts [at O(S 0)] in H4 ≡ H4,2−2 + H4,3−1 + H4,1−3.
+The quartic part has been decomposed into a 2 − 2 scattering
+term, H4,2−2, and anomalous 3−1 and 1−3 terms, H4,3−1 and
+H4,1−3. Since J0,±
+k
+= 0, there are no three boson terms in H
+[Eq. (S4)]. In terms of the local exchanges, the coefficents in
+H2 and H4 are given explicitly by
+Ak = S
+�
+J+−
+k
+− J00
+0
+�
+,
+Bk = S J++
+k ,
+Vk,k′,q = 1
+2
+�
+J00
+k′−k−q + J00
+−q + J00
++q + J00
+k−k′+q
+�
+− 1
+2
+�
+J+−
+k
++ J+−
+k′ + J+−
+k′−q + J+−
+k+q
+�
+,
+Dk,k′,q = −1
+2
+�
+J++
+k
++ J++
+k′ + J++
+q
+�
+.
+By construction, these coefficients must satisfy the symmetry
+relations
+Ak = A∗
+k,
+Bk = B−k,
+Vk,k′,q = Vk′,k,−q = Vk,k′,k′−k−q = Vk′,k,k−k′+q = V∗
+k+q,k′−q,−q,
+Dk,k′,q = Dk,q,k′ = Dk′,k,q = Dk′,q,k = Dq,k,k′ = Dq,k′,k.
+A.
+Non-Interacting Spin-Waves
+Consider first only the quadratic (non-interacting magnon)
+portion of H,
+H2 =
+�
+k
+�
+Aka†
+kak + 1
+2!
+�
+Bka†
+ka†
+−k + H.c.
+��
+.
+(S6)
+
+11
+4
+This can be diagonalized by the usual Bogoliubov transforma-
+tion [8]. Defining the matrix
+Mk ≡
+�
+Ak Bk
+B∗
+k Ak
+�
+,
+(S7)
+the spin-wave spectrum is obtained by diagonalization of
+σzMk, where σz is a (block) Pauli matrix. One finds the pos-
+itive frequency mode
+ωk =
+�
+A2
+k − |Bk|2 > 0.
+For the ferromagnetic Heisenberg-compass model, Ak and Bk
+are given by
+Ak = −S
+��
+2J + Ksin2ϕ
+�
+cos kx +
+�
+2J + Kcos2ϕ
+�
+cos ky −
+2(2J + K)
+�
+Bk = −S
+��
+Ksin2ϕ
+�
+cos kx +
+�
+Kcos2ϕ
+�
+cos ky
+�
+.
+Note that A0 = KS and B0 = −KS , yielding a zero energy
+mode at k = 0, with ω0 = 0 and with both Ak and Bk real.
+The eigenvector of σzMk associated with the positive mode
+can be written as (uk, vk) where
+uk =
+�
+ωk + Ak
+2ωk
+,
+vk = −
+Bk
+√2ωk(ωk + Ak)
+,
+which we have defined so that u2
+k − v2
+k = 1. Note that since
+both Ak and Bk are inversion even, we have u−k = uk, v−k = vk
+and ωk = ω−k. Since both Ak and Bk are real, we find that uk
+and vk are real as well. The diagonalized boson operators are
+defined via
+ak = ukγk + vkγ†
+−k,
+a†
+k = vkγ−k + ukγ†
+k.
+Expectation values of bilinears of these bosons can be written
+in terms of uk and vk. Noting that at temperature T these are
+⟨γ†
+kγk⟩ = nB(ωk),
+⟨γkγ†
+k⟩ = 1 + nB(ωk),
+where nB(ω) = [exp(ω/T)−1]−1 is the boson thermal occupa-
+tion number. The above thermal expectations for the original
+a-bosons are given by
+⟨a†
+kak⟩ = nB(ωk)u2
+k + [1 + nB(ωk)] v2
+k,
+⟨aka−k⟩ = ⟨a†
+−ka†
+k⟩
+∗ = [1 + 2nB(ωk)] ukvk.
+In the classical limit, where T ≫ ωk, we have nB(ωk) ≈
+T/ωk ≫ 1. The expectations then become
+⟨a†
+kak⟩ = T
+ωk
+�
+u2
+k + v2
+k
+�
+= T
+ωk
+� Ak
+ωk
+�
+,
+(S8a)
+⟨aka−k⟩ = ⟨a†
+−ka†
+k⟩ = 2T
+ωk
+ukvk = − T
+ωk
+� Bk
+ωk
+�
+.
+(S8b)
+Finally, the ordered moment (selected by ObTD), M ≡
+1
+N
+�
+r⟨Sr⟩ ≡ M ˆx, can be expressed in terms of these boson
+averages as
+M = S − 1
+N
+�
+k
+⟨a†
+kak⟩ ≡ S
+1 − T
+S N
+�
+k
+Ak
+ω2
+k
+ .
+(S9)
+B.
+Interacting Spin-Waves
+To consider the effects of the quartic parts of H in Eq. S4,
+H4,2−2, H4,3−1 and H4,1−3, we adopt a mean-field like ap-
+proach, replacing each possible contraction of operators with
+averages with respect to the quadratic, or “free” part, H2 [10,
+11]. This procedure is equivalent to leading order perturbation
+theory in the interactions [12, 13]. For example, consider the
+scattering term
+a†
+k+qa†
+k′−qak′ak ≈ ⟨a†
+k+qak′⟩a†
+k′−qak + ⟨a†
+k′−qak⟩a†
+k+qak′
++ ⟨a†
+k+qak⟩a†
+k′−qak′ + ⟨a†
+k′−qak′⟩a†
+k+qak
++ ⟨a†
+k+qa†
+k′−q⟩ak′ak + ⟨ak′ak⟩a†
+k+qa†
+k′−q.
+Using that the expectation values satisfy ⟨a†
+kak′⟩ ∝ δk,k′ and
+⟨akak′⟩ ∝ δk,−k′ one finds
+a†
+k+qa†
+k′−qak′ak ≈
+�
+δq,0 + δk+q,k′
+� �
+⟨a†
+k′ak′⟩a†
+kak + ⟨a†
+kak⟩a†
+k′ak′
+�
++ δk,−k′
+�
+⟨a†
+k+qa†
+−k−q⟩a−kak + ⟨a−kak⟩a†
+k+qa†
+−k−q
+�
+.
+Combing this decomposition with the interaction vertex, as
+specified in Eq. (S5b), gives the expression
+H4,2−2 ≈
+�
+k
+
+1
+N
+�
+q
+Vk,q,0⟨a†
+qaq⟩
+ a†
+kak
++1
+2
+�
+k
+
+
+1
+2N
+�
+q
+Vq,−q,k−q⟨aqa−q⟩
+ a†
+ka†
+−k + H.c.
+ ,
+where Vk,k′,k′−k = Vk,k′,0 and Vk,k′,0 = Vk′,k,0 has been used
+to simplify the normal term, and shifting the momentum has
+been used to simplify the anomalous terms. The quartic terms
+thus appear as corrections to the Ak and Bk quadratic terms.
+Next, consider the same manipulations for the anomalous
+boson terms, starting with
+a†
+ka†
+k′a†
+qak+k′+q ≈ ⟨a†
+ka†
+k′⟩a†
+qak+k′+q + a†
+ka†
+k′⟨a†
+qak+k′+q⟩
++ ⟨a†
+ka†
+q⟩a†
+k′ak+k′+q + a†
+ka†
+q⟨a†
+k′ak+k′+q⟩
++ ⟨a†
+kak+k′+q⟩a†
+k′a†
+q + a†
+kak+k′+q⟨a†
+k′a†
+q⟩.
+Using the fact that the expectations in this last equation are
+diagonal in k (or skew-diagonal) [as in Eq. (S8)], we find
+a†
+ka†
+k′a†
+qak+k′+q ≈ δk,−k′
+�
+⟨a†
+ka†
+−k⟩a†
+qaq + a†
+ka†
+−k⟨a†
+qaq⟩
+�
++ δk,−q
+�
+⟨a†
+ka†
+−k⟩a†
+k′ak′ + a†
+ka†
+−k⟨a†
+k′ak′⟩
+�
++ δk′,−q
+�
+⟨a†
+kak⟩a†
+k′a†
+−k′ + a†
+kak⟨a†
+k′a†
+−k′⟩
+�
+.
+Combining this decomposition with the anomalous interac-
+tion vertex, Dk,k′,q from Eq. (S5c), and using the permutation
+symmetry of its arguments, we find
+H4,3−1 ≈
+�
+k
+
+1
+2N
+�
+q
+Dq,−q,k⟨a†
+qa†
+−q⟩
+ a†
+kak
++1
+2
+�
+k
+
+1
+N
+�
+q
+Dk,−k,q⟨a†
+qaq⟩
+ a†
+ka†
+−k.
+
+12
+5
+These terms thus also appear as corrections to the Ak and Bk
+in the quadratic part of the Hamiltonian. Note that the Her-
+mitian conjugate term of this H4,3−1 also contributes, with its
+contribution read off from the expression above.
+H4,1−3 ≈
+�
+k
+
+1
+2N
+�
+q
+D∗
+q,−q,k⟨aqa−q⟩
+ a†
+kak
++1
+2
+�
+k
+
+1
+N
+�
+q
+D∗
+k,−k,q⟨a†
+qaq⟩
+ a−kak.
+Finally, we can summarize all of these contributions as cor-
+rections δAk and δBk to the original Ak and Bk of quadratic
+H2 origin and write
+δAk = 1
+N
+�
+q
+�
+Vk,q,0⟨a†
+qaq⟩ + 1
+2
+�
+Dq,−q,k⟨a†
+qa†
+−q⟩ + c.c.
+��
+,
+(S10a)
+δBk = 1
+N
+�
+q
+�
+Dk,−k,q⟨a†
+qaq⟩ + 1
+2Vq,−q,k−q⟨aqa−q⟩
+�
+.
+(S10b)
+In terms of these corrections, the renormalized spectrum is
+given by
+Ωk ≡
+�
+(Ak + δAk)2 − (Bk + δBk)2.
+(S11)
+These corrections can be evaluated using the bare, free av-
+erages from Eq. (S8), though this approach leads to diver-
+gences (see Sec. III F). Alternatively, they can be evaluated
+self-consistently, with the averages in Eq. (S8) computed us-
+ing (Ak + δAk), (Bk + δBk) and Ωk instead of Ak, Bk and ωk,
+which cures the divergences.
+C.
+Pseudo-Goldstone gap
+The effects of the interactions on the pseudo-Goldstone
+mode can now be examined. The energy of the k = 0 mode is
+given by
+∆ ≡ Ω0 =
+�
+2KS (δA0 + δB0) + δA2
+0 − δB2
+0.
+(S12)
+For small corrections δA0, δB0, ∆ above can be approxi-
+mated by (the leading term)
+∆ ≈
+√
+2KS
+�
+δA0 + δB0.
+(S13)
+In the quantum limit where T ≪ ωk, the corrections δAk,
+δBk are O(S 0) and thus the gap scales as ∆ ∝
+√
+S .
+In
+the classical limit where T ≫ ωk the corrections scale as
+δAk, δBk ∼ O(T/S ) and thus the gap scales as ∆ ∝
+√
+T, inde-
+pendent of S .
+D.
+Pseudo-Goldstone Linewidth
+To estimate the scaling of the pseudo-Goldstone mode
+linewidth with temperature, we consider the magnon self-
+energy [11] at k = 0 near ω = 0, which takes the form
+Σ(0, 0) ≡
+�
+δA0 δB0
+δB∗
+0 δA∗
+0
+�
+,
+where δA0 and δB0 are corrections due to magnon-magnon
+interactions. Perturbatively, we expect that
+δA0 = a1T + a2T 2 + · · · ,
+(S14a)
+δB0 = b1T + b2T 2 + · · · ,
+(S14b)
+where the O(T) corrections (computed in this work) encoded
+in a1, b1 are both real. The quasi-normal modes, correspond-
+ing to the locations of poles of the magnon Green’s func-
+tion [11, 14], are determined from eigenvalues of σzMeff
+0
+where
+Meff
+0 =
+�
+A0 + δA0
+−A0 + δB0
+−A0 + δB∗
+0
+A0 + δA∗
+0
+�
+.
+Up to and including terms of O(T 2), the quasi-normal mode
+frequency is thus given by
+Re Ω0 ≈
+�
+2A0(a1 + b1)
+√
+T
++
+
+a2
+1 − b2
+1 + 2A0(Re a2 + Re b2)
+4A0(a1 + b1)
+ T 3/2 + · · · ,
+Im Ω0 ≈ (Im a2)T 2 + · · · .
+We thus see that the linewidth, determined by Im Ω0, is ex-
+pected to scale as T 2.
+E.
+Self-Consistent Mean-Field Theory (SCMFT)
+To include the effects of the magnon-magnon interactions
+self-consistently, we define the “mean-fields”
+nk ≡ ⟨a†
+kak⟩,
+dk ≡ ⟨a†
+ka†
+−k⟩.
+(S15)
+Using Eq. (S10), new values of nk and dk can then be com-
+puted by iteratively updating Ak and Bk to
+A′
+k = Ak + 1
+N
+�
+q
+�
+Vk,q,0nq + 1
+2
+�
+Dq,−q,kdq + D∗
+q,−q,kd∗
+q
+��
+,
+B′
+k = Bk + 1
+N
+�
+q
+�
+Dk,−k,qnq + 1
+2Vq,−q,k−qd∗
+q
+�
+,
+which, using Eq. (S8), results in updated values of nk and dk.
+This process is repeated until the variables nk and dk have con-
+verged to the desired precision across the full Brillouin zone.
+For the calculations reported here, and in the main text, con-
+vergence was considered reached when the sum of all absolute
+values of the changes in nk and dk in Eq. (S15) over the Bril-
+louin zone between iterations was less than 10−10. To launch
+the iterative process, the mean-fields, nk and dk for each k,
+are initially set to a value of 1/2, though the precise choice of
+initial value was not found to affect the final results. Follow-
+ing this approach, we calculate the PG gap for several system
+
+13
+6
+sizes, using a discrete sum of the Brillouin zone with N = L2
+points. We then extrapolate the gap in the system size to ob-
+tain the result in the thermodynamic limit (N → ∞). The
+finite size scaling of the PG gap using SCMFT is shown in
+Fig. S2.
+F.
+Cancellation of divergences in the pseudo-Goldstone gap
+Since the non-interacting LSWT spectrum is gapless, we
+must be mindful of infrared divergent contributions to δA0 and
+δB0. Let us first address this issue in the simplest context, bare
+perturbation theory in the quartic interactions.
+We focus on the classical limit where ωk ≪ T, but similar
+considerations apply in the full quantum case at finite tem-
+perature; since ωk → 0 as k → 0, there is always a regime
+in k near the zone center where the frequency is small rela-
+tive to temperature, even in the quantum limit. Consider the
+corrections, Eq. (S10), in the thermodynamic limit (N → ∞),
+replacing the discrete sums with integrals. At k = 0, this gives
+[using Eq. (S8)]
+δA0 =
+�
+d2q
+(2π)2
+T
+ω2q
+�
+V0,q,0Aq − Dq,−q,0Bq
+�
+,
+(S16a)
+δB0 =
+�
+d2q
+(2π)2
+T
+ω2q
+�
+D0,0,qAq − 1
+2Vq,−q,−qBq
+�
+,
+(S16b)
+where the integral is over the Brillouin zone −π ≤ qx, qy ≤
+π (the lattice spacing has been set to one). At small q, the
+spectrum is approximately linear in q with
+ωq = S
+�
+2K
+�
+J|q|2 + Kq2y
+�
++ O(|q|2)
+and thus the factor T/ω2
+q ∝ 1/|q|2 is singular as |q| → 0. The
+numerators of the integrals in Eq. (S16) remain finite in this
+limit, with
+V0,q,0Aq − Dq,−q,0Bq = −S K2
+2
++ O(|q|2),
+D0,0,qAq − 1
+2Vq,−q,−qBq = +S K2
+2
++ O(|q|2).
+One therefore finds that both δA0 and δB0 are logarithmically
+divergent. Explicitly, integrating over a region 2π/L < |q| <
+Λ ≪ π
+�
+2π/L<|q|<Λ
+d2q
+(2π)2
+1
+ω2q
+=
+1
+4πS 2K √J(J + K)
+ln
+�LΛ
+2π
+�
+.
+(S17)
+Since the upper cutoff is chosen to satisfy Λ ≪ π, the diver-
+gent contributions to δA0 and δB0 take the form
+δA0 = −
+TK ln L
+8πS √J(J + K)
++ (reg.),
+(S18a)
+δB0 = +
+TK ln L
+8πS √J(J + K)
++ (reg.),
+(S18b)
+where (reg.) stands for terms that remain finite as L → ∞. In-
+terestingly, while δA0 and δB0 are each ln L divergent, the sum
+(δA0 + δB0) which appears in the expression for the pseudo-
+Goldstone gap [Eq. (S13)], ∆, is finite. This can be made more
+explicit by carrying out the same expansions for (δA0 + δB0),
+δA0 + δB0 =
+�
+d2q
+(2π)2
+T
+ω2q
+�
+2K2S (q2
+x − q2
+y) + O(|q|4)
+�
+. (S19)
+The O(|q|2) term in the ω2
+q denominator is thus com-
+pensated by a corresponding O(|q|2) in the numerator of
+Eq. (S19).
+However, note that this cancellation only oc-
+curs at leading order in δA0, δB0. The complete expression
+�
+(A0 + δA0)2 − (B0 + δB0)2, which incorporates higher-order
+contributions, remains logarithmically divergent. Similarly,
+the leading corrections from bare perturbation theory to Ωk at
+non-zero k are also divergent.
+Since the bare perturbation theory diverges, except for the
+leading temperature dependence of the PG gap at q = 0, in
+order to obtain the full temperature dependence of the inter-
+action corrections to Ωq, we proceed with a self-consistent
+approach. This way, the equation for the corrections δA0 and
+δB0 become
+δA0 =
+�
+d2q
+(2π)2
+T
+Ω2q
+�
+V0,q,0(Aq + δAq) − Dq,−q,0(Bq + δBq)
+�
+,
+δB0 =
+�
+d2q
+(2π)2
+T
+Ω2q
+�
+D0,0,q(Aq + δAq) − 1
+2Vq,−q,−q(Bq + δBq)
+�
+,
+where the renormalized spectrum Ωq arises from evaluating
+the averages in Eq. (S8) self-consistently.
+In such a self-
+consistent mean-field theory, the spectrum Ωq “already” con-
+tains a finite gap at q = 0. The gap acts as an effective infrared
+cutoff rendering the integrals in Eq. (S17) finite. The disap-
+pearance of the divergence then manifests itself in the cancel-
+lation of the leading (self-consistent) dependence on the gap
+∆.
+To see this explicitly, consider the self-consistent spectrum
+which, for small q, takes the form
+Ωq =
+�
+2KS 2 �
+J|q|2 + Kq2y
+�
++ ∆2 + O(|q|2).
+For sufficiently small ∆, the integration region can be divided
+into two parts: |q| ≳ k0 and |q| ≲ k0 such that
+Ωq ≈
+
+∆,
+|q| ≲ k0,
+S
+�
+2K
+�
+J|q|2 + Kq2y
+�
+,
+|q| ≳ k0.
+Roughly, the boundary separating these regions scales as
+k0 ∼ ∆
+S K ∝
+√
+T
+(S21)
+when K ≳ J. Alternatively, k0 is the wave-vector at which
+the bare spectrum ωk becomes comparable to the interaction
+induced gap, ∆. The primary change to the spectrum, and thus
+to δA0 and δB0, occurs for |q| < k0. Carrying out the integra-
+tion in Eq. (S17) over the region responsible for its divergent
+contributions, we find they are rendered finite. Explicitly,
+�
+k0<|q|<Λ
+d2q
+(2π)2
+1
+Ω2q
+∼
+ln (KS Λ/∆)
+4πS 2K √J(J + K)
+.
+
+14
+7
+Given that ∆ ∝
+√
+T, this contribution to the integral now
+scales as − ln T. Thus the divergence has been cured in the in-
+dividual corrections δA0 and δB0. We note that the ln(Λ/∆) ∼
+− ln T terms cancel in the sum (δA0 + δB0) which controls the
+leading contribution to the gap [similarly to Eq. (S13)] and
+the result from bare perturbation theory is recovered. In this
+way, bare perturbation theory for the asymptotic
+√
+T scaling
+of the pseudo-Goldstone gap is well-defined and divergence
+free, and matches the results from the SCMFT calculations.
+Note that the region 0 < |q| < k0 gives only a finite contribu-
+tion that goes as
+�
+0<|q|<k0
+d2q
+(2π)2
+1
+Ω2q
+∼
+k2
+0
+4π∆2 ∼ const. ,
+since k0 ∝ ∆.
+G.
+Logarithmic corrections to magnetization
+While the contributions ∝ ln T cancel in the leading parts
+of ∆, they reappear explicitly in static quantities such as the
+magnetization. In the classical limit, the net magnetization
+[Eq. (S9)] is given by
+M = S − T
+�
+d2k
+(2π)2
+Ak
+ω2
+k
+,
+(S22)
+in the thermodynamic limit (N → ∞). Like the bare cor-
+rections δA0 and δB0 in Eq. (S16), in LSWT M has a (log-
+arithmic) infrared divergence, given that ωk scales linearly
+with k at small |k|, while Ak tends to the constant A0 = KS .
+Na¨ıvely, this would indicate the destruction of the order, as
+happens when there is a true symmetry-protected Goldstone
+mode [15].
+To resolve this divergence, we must include the dynami-
+cally generated pseudo-Goldstone gap. The expression for M
+in the SCMFT can be obtained from Eq. (S22), via the re-
+placements ωk → Ωk and Ak → Ak + δAk. Explicitly,
+M = S − T
+�
+d2k
+(2π)2
+Ak + δAk
+Ω2
+k
+,
+(S23)
+where, again, Ωk is the self-consistent spectrum with pseudo-
+Goldstone gap ∆. Following the same strategy as in Sec. III F,
+we approximate the self-consistent spectrum, Ωk, over the
+three pertinent regions of reciprocal space
+Ωk ≈
+
+∆,
+0 < |k| ≲ k0
+S
+�
+2K
+�
+J|k|2 + Kk2y
+�
+,
+k0 ≲ |k| ≲ Λ
+ωk.
+|k| ≳ Λ
+(S24)
+The integral defining M is then split into three parts
+�
+d2k
+(2π)2 =
+�
+0<|k|<k0
+d2k
+(2π)2 +
+�
+k0<|k|<Λ
+d2k
+(2π)2 +
+�
+|k|>Λ
+d2k
+(2π)2 .
+The first and second integrals depend on temperature through
+∆ ∝
+√
+T and k0 ∼ ∆/(KS ) [see Eq. (S21)]. The last integral
+is over wave-vectors large enough such that the interaction
+corrections are minor and, therefore, this contribution to the
+integral has no additional temperature dependence. The cor-
+rection to M from this last (third) term is thus ∝ T.
+For the region |k| ≲ Λ, we approximate Ak + δAk ≈ KS ,
+leaving the two contributions
+�
+0<|k|<k0
+d2k
+(2π)2
+KS T
+∆2
+=
+T
+4πKS ,
+�
+k0<|k|<Λ
+d2k
+(2π)2
+KS T
+2KS 2 �
+J|k|2 + Kk2y
+� = T ln (KS Λ/∆)
+4πS √J(J + K)
+.
+The renormalized spectrum has thus yielded two additional
+contributions to M: one that adds an additional part ∝ T and
+another new, and perhaps most interesting, part, ∝ T ln T, and
+where we used ∆ ∝
+√
+T.
+To summarize, the magnetization at low temperatures takes
+the form
+M = S − c1T − c2T ln T + · · · ,
+(S25)
+where c1 and c2 are temperature independent constants. The
+logarithmic T ln T dependence arises from the temperature
+dependence of the pseudo-Goldstone gap, ∆. As T → 0,
+both of these temperature dependent terms go to zero (as it
+should classically) and the system becomes fully polarized
+with M → S . Hence true long-range order is induced by
+ObTD, with the lurking infrared divergences tamed by the
+ObTD-induced PG gap, ∆.
+[1] J. D. Alzate-Cardona, D. Sabogal-Su´arez, R. F. L. Evans, and
+E. Restrepo-Parra, “Optimal phase space sampling for Monte
+Carlo simulations of Heisenberg spin systems,” Journal of
+Physics: Condensed Matter 31, 095802 (2019).
+[2] Michael Creutz, “Overrelaxation and Monte Carlo simulation,”
+Phys. Rev. D 36, 515–519 (1987).
+[3] M. E. J. Newman and G. T. Barkema, Monte Carlo methods in
+statistical physics (Clarendon Press, Oxford, 1999).
+[4] J. R. Dormand and P. J. Prince, “A family of embedded Runge-
+Kutta formulae,” Journal of Computational and Applied Math-
+ematics 6, 19–26 (1980).
+[5] K. Ahnert and M. Mulansky, “Odeint – solving ordinary differ-
+ential equations in C++,” AIP Conference Proceedings 1389,
+1586–1589 (2011).
+[6] K. Ahnert and M. Mulansky, “Boost C++ Library: Odeint,”
+(2012).
+[7] J. C. Bowman and M. Roberts, “FFTW++: A fast Fourier trans-
+form C++ header class for the FFTW3 library,” (2010).
+[8] A. Auerbach, Interacting Electrons and Quantum Magnetism,
+Graduate Texts in Contemporary Physics (Springer New York,
+1998).
+[9] Jeffrey G. Rau, Paul A. McClarty,
+and Roderich Moess-
+
+15
+8
+ner, “Pseudo-Goldstone gaps and order-by-quantum disorder in
+frustrated magnets,” Phys. Rev. Lett. 121, 237201 (2018).
+[10] H. Bruus and K. Flensberg, Many-Body Quantum Theory in
+Condensed Matter Physics: An Introduction, Oxford Graduate
+Texts (Oxford University Press, Oxford, 2004).
+[11] J.P. Blaizot and G. Ripka, Quantum Theory of Finite Systems
+(MIT Press, Cambridge, 1986).
+[12] P. D. Loly, “The Heisenberg ferromagnet in the selfconsistently
+renormalized spin wave approximation,” Journal of Physics C:
+Solid State Physics 4, 1365–1377 (1971).
+[13] A. V. Chubukov, S. Sachdev, and T. Senthil, “Large-S expan-
+sion for quantum antiferromagnets on a triangular lattice,” Jour-
+nal of Physics: Condensed Matter 6, 8891–8902 (1994).
+[14] M. E. Zhitomirsky and A. L. Chernyshev, “Colloquium: Spon-
+taneous magnon decays,” Rev. Mod. Phys. 85, 219–242 (2013).
+[15] P. M. Chaikin and T. C. Lubensky, Principles of Condensed
+Matter Physics (Cambridge University Press, 1995).
+

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@@ -0,0 +1,688 @@
+arXiv:2301.05072v1  [cond-mat.stat-mech]  12 Jan 2023
+Chemical kinetics and stochastic differential equations
+Chiara Pezzotti and Massimiliano Giona1, ∗
+1Dipartimento di Ingegneria Chimica, Materiali,
+Ambiente La Sapienza Universit`a di Roma
+Via Eudossiana 18, 00184 Roma, Italy
+(Dated: January 13, 2023)
+Abstract
+We propose a general stochastic formalism for describing the evolution of chemical reactions
+involving a finite number of molecules. This approach is consistent with the statistical analysis
+based on the Chemical Master Equation, and provides the formal setting for the existing algorithmic
+approaches (Gillespie algorithm). Some practical advantages of this formulation are addressed, and
+several examples are discussed pointing out the connection with quantum transitions (radiative
+interactions).
+∗ corresponding author:[email protected]
+1
+
+All the chemical physical processes involve, in an atomistic perspective, a stochastic de-
+scription of the events, be them reactive or associated with a change of phase (for instance
+adsorption) [1]. Nonetheless, in the overwhelming majority of the cases of practical and labo-
+ratory interest, the number of molecules involved is so large to justify a mean field approach,
+essentially based on the Boltzmannian hypothesis of molecular chaos (the “stosszahlansatz”)
+[2]. The mean field formulation represents the backbone of the classical theory of chemical
+reaction kinetics [3, 4].
+It is well known that, in all the cases where the number of molecule is small (and this
+occurs in subcellular biochemical reactions, in nanoscale systems, or in the growth kinetics of
+microorganisms [5–7]), the effects of fluctuations become significant, motivating a stochastic
+description of chemical kinetic processes, involving the number of molecules present in the
+system, thus explicitly accounting for due to their finite number [8–11]. The statistical theory
+of chemical kinetics in these conditions is grounded on the Chemical Master Equation (CME)
+[12, 13], expressing the evolution equation for the probabilities p(N, t) of all the possible
+number-configurations N(t) = (N1(t), . . . , Ns(t)), where Nh(t) is the number of molecules of
+the h-th reacting species at time t, h = 1, . . . , s. However, apart from a handful of simple
+cases, for which the CME can be solved analytically [14], numerical methods should be
+applied to it in order to compute mean values and higher-order moments. But also this
+choice reveals itself to be unfeasible in most of the situations of practical and theoretical
+interests, due to the extremely large number of configurations involved, making the multi-
+index matrix p(N, t) so huge to exceed reasonable computational facilities.
+In order to solve this problem, Gillespie proposed an algorithmic solution to the numeri-
+cal simulation of stochastic reacting systems, based on the Markovian nature of the reactive
+events [15, 16]. The original Gillespie algorithm has been extended and improved over time,
+providing a variety of slightly different computational alternatives. A common denominator
+of the first family of the Gillespie algorithms (namely those based on the direct method, the
+first reaction method or their derivates [17–19]) is to associate to every time step the occur-
+rence of just one reaction. This formulation comes directly from the assumption that, if the
+time step is small enough, the probability that more than one reaction will occur is negligi-
+ble. While correct, this choice brings to significant computational costs for complex reaction
+schemes. This problem has been highlighted several times, from the Gillespie group itself,
+as stiffness in stochastic chemical reacting systems [20]. A brilliant way to overcome this
+2
+
+limit originates the famous tau-leaping method, which, unfortunately, requires to check that
+the propensity functions remain almost constant at each iteration and can be applied just
+if this condition is verified [21, 22]. The algorithmic solution associated with the formalism
+here introduced combines the accuracy of the first SSA with the computational advantages
+of the τ-leaping method.
+There is, moreover, a missing link between the CME theory and the Gillespie algorithm,
+consisting in the straight mathematical formulation of the stochastic differential equations
+associated with a chemical reacting system, the statistical description of which would cor-
+respond to the CME. To clarify this issue, consider the conceptually analogous problem of
+particle diffusion over the real line, the statistical description of which is expressed by the
+parabolic equation ∂p(x, t)/∂t = D ∂2p(x, t)/∂x2, for the probability density p(x, t) of finding
+a particle at position x at time t. Setting xn = x(n ∆t), an algorithm describing this process
+can be simply expressed by the discrete evolution equation xn+1 = xn +
+√
+2 D ∆t rn+1, where
+rh, h = 1, 2, . . . represent independent random variables sampled from a normal distribution
+(with zero mean, and unit variance) [23]. This represents an efficient algorithmic solution
+of the problem, whenever the time resolution ∆t is small enough. Nevertheless, the mere
+algorithmic approach cannot be considered physically satisfactory, in a comprehensive for-
+mulation of transport theory embedded in a continuous space-time (in which both position x
+and time t are real valued). In point of fact, only with the mathematical formulation due to
+K. Ito of stochastic differential equations driven by the increments dw(t) of a Wiener process
+(Langevin equations) [24], namely dx(t) =
+√
+2 D dw(t) the theory of diffusive motion has
+found a proper mathematical physical setting.
+A similar situation applies to the case of stochastic models of chemical reaction kinetics,
+and the present Letter is aimed at filling this gap.
+The basic idea is that any reactive
+process corresponds to a system of elementary events (the single reaction) possessing a
+Markovian transitional structure, and, consequently, amenable to a description by means
+of the increments of counting processes (Poisson processes, in the Markovian case). This
+topic has been also pointed out in [25] in terms of Poisson measures, although the latter
+formulation is much less simple and physically intuitive than the approach proposed in the
+present Letter.
+To begin with, consider the simple case of a first-order chemical reaction A
+k1
+⇋
+k−1 B (for
+instance, an isomerization). This model is perfectly analogous to the radiative transition
+3
+
+FIG. 1. Schematic representation of the analogy between a two-level quantum system and a first-
+order chemical kinetics, such as an isomerization.
+of a molecule possessing two energy states, due to emission and adsorption of an energy
+quantum (figure 1). Let NA(0) + NB(0) = Ng the total number of molecules at time t = 0.
+The state of the system is characterized by the state functions σh(t), h = 1, . . . , Ng for each
+molecule, attaining values {0, 1}, and such that σh(t) = 0 if the energy state at time t is E0
+(or equivalently if the molecule finds itself in the state A), and σh(t) = 1 in the opposite
+case (energy state E1, or isomeric state B).
+Let {χ(1)
+h (t, k1), χ(2)
+h (t, k−1)}Ng
+h=1 be two systems of independent Poisson processes, char-
+acterized by the transition rates k1, and k−1, respectively. The evolution of σh(t) can be
+expressed via the stochastic differential equation
+dσh(t)
+dt
+= (1 − σh(t)) dχ(1)
+h (t, k1)
+dt
+− σh(t) dχ(2)
+h (t, k−1)
+dt
+(1)
+h = 1, . . . , Ng, where dχ(t, λ)/dt is the distributional derivative of the Poisson process
+χ(t, λ), corresponding to a sequence of unit impulsive functions at the transition instants
+t∗
+i , i = 1, 2, . . . , 0 < t∗
+i < t∗
+i+1, where for ε > 0, limε→0
+� t∗
+i +ε
+t∗
+i −ε dχ(t, λ) = 1. Summing over
+h = 1, . . . Ng, and observing that NA(t) = �Ng
+h=1 (1 − σh(t)), NB(t) = �Ng
+h=1 σh(t), we have
+dNB(t)
+dt
+=
+NA(t)
+�
+h=1
+dχ(1)
+h (t, k1)
+dt
+−
+NB(t)
+�
+h=1
+dχ(2)
+h (t, k−1)
+dt
+(2)
+4
+
+(b)(a)OPand dNA(t)/dt = −dNB(t)/dt, representing the evolution equation for NA(t) and NB(t),
+attaining integer values. The stochastic evolution of the number of molecules NA(t), NB(t)
+is thus expressed as a differential equation with respect to the continuous physical time
+t ∈ R+, over the increments of a Poisson process. Intepreted in a mean-field way, if ctot is
+the overall concentration of the reactants at time t = 0, then the concentrations cα(t) at
+time t can be recovered from eq. (2) as
+cα(t) = ctot
+Nα(t)
+Ng
+,
+α = A, B
+(3)
+representing the calibration relation connecting the stochastic description in terms of num-
+ber of molecules Nα(t) and the concentrations cα(t), α = A, B entering the mean-field
+description.
+The analytical formulation of a stochastic differential equation for chemical kinetics,
+expressed in terms of the number of molecules of the chemical species involved, rather than
+an algorithm defined for discretized times, permits to develop a variety of different numerical
+strategies, that naturally perform a modified tau-leaping procedure, as the occurrence of
+several distinct reactive events in any elementary time step ∆t is intrinsically accounted for.
+This can be easily seen by considering the simple reaction defined by the evolution equation
+(2).
+In terms of increments, eq.
+(2) can be written as dNB(t) = �NA(t)
+h=1 dχ(1)(t, k1) −
+�NB(t)
+h=1 dχ(2)(t, k−1). If ∆t is the chosen time step, it follows from this formulation, a simple
+numerical approximation for eq. (2), namely,
+∆NB(t) = NB(t + ∆t) − NB(t) =
+NA(t)
+�
+h=1
+ξ(1)
+h (k1 ∆t) −
+NB(t)
+�
+h=1
+ξ(2)
+h (k−1 ∆t)
+(4)
+where ξ(1)(k1 ∆t), ξ(2)
+h (k−1 ∆t) h = 1, 2, . . . , are two families of independent binary random
+variables, where
+ξ(α)
+h (p) =
+
+
+
+1
+with probability p
+0
+otherwise
+(5)
+α = 1, 2, h = 1, 2, . . . . The time step ∆t, can be chosen in eq. (4) from the condition
+K ∆t < 1 ,
+K = max{k1, k−1}
+(6)
+In practice, we choose ∆t = 0.1/K. As can be observed, the choice of ∆t is limited by the
+intrinsic rates of the process. The advantage of deriving different algorithmic schemes for
+5
+
+solving numerically the stochastic kinetic equations becomes more evident in dealing with
+bimolecular reactions (addressed below). Due to the intrinsic limitations of this commu-
+nication, a thorouh discussion of this issue is postponed to a future more extensive article
+[26].
+The same approach can be extended to include amongst the elementary events not only
+the reactive steps, but also feeding conditions, thus representing the evolution of chemically
+reacting systems with a finite number of molecules in a perfectly stirred open reactor. This is
+the case of the tank-loading problem, in which a tracer is injected in an open vessel assumed
+perfectly mixed, for which, in the absence of chemical reactions, the mean field equation for
+the concentration of the tracer reads
+dc(t)
+dt
+= D (c0 − c(t))
+(7)
+where c0 is the inlet concentration and D the dilution rate (reciprocal of the mean retention
+time), and c(0) = 0. Fixing Ng so that c(t) = c0 N(t)/Ng, the corresponding stochastic
+differential equation for the integer N(t) involves, also in this case, two families of counting
+processes, one for the loading at constant concentration c0, and the other for tracer discharge
+in the outlet stream, characterized by the same transition rate D,
+dN(t)
+dt
+=
+Ng
+�
+h=1
+dχ(1)
+h (t, D)
+dt
+−
+N(t)
+�
+k=1
+dχ(2)
+h (t, D)
+dt
+(8)
+starting from N(0) = 0. Figure 2 depicts several realizations of the tank-loading process,
+obtained by discretizing eq. (8) with a time step ∆t = 10−3. Despite the simplicity of the
+process, this example permits to highlight the role of Ng, that can be referred to as the granu-
+larity number, and the way stochastic models of chemical reactions can be fruitfully applied.
+Indeed, there is a two-fold use of the stochastic formulation of chemical kinetic schemes.
+The first refers to a chemical reacting system involving a small number of molecules, and
+in this case Ng represents the effective number of molecules present in the system. The
+other is to use stochastic algorithms for simulating reacting systems in an alternative (and
+sometimes more efficient way) with respect to the solution of the corresponding mean-field
+equations. In the latter case, the granularity number Ng represents essentially a computa-
+tional parameter, tuning the intensity of the fluctuations. Two choices are then possible:
+(i) it can be chosen large enough, in order to obtain from a single realization of the process
+an accurate approximation of the mean-field behavior, or (ii) it can be chosen small enough
+6
+
+ 0
+ 0.4
+ 0.8
+ 1.2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+c(t)
+t
+ 0
+ 0.4
+ 0.8
+ 1.2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+c(t)
+t
+ 0
+ 0.4
+ 0.8
+ 1.2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+c(t)
+t
+(a)
+(b)
+(c)
+FIG. 2. c(t) = N(t)/Ng vs t from a single realization of the tank-loading process eq. (8) with
+D = 1, c0 = 1 a.u.. Panel (a): Ng = 30, panel (b) Ng = 100, panel (c) Ng = 1000. The solid
+horizontal lines represent the steady-state value c∗ = 1.
+in order, to deal with extremely fast simulations of a single realization of the process, that
+could be averaged over a statistically significant number of realizations in due time. These
+two choices are depicted in figure 2 (panel c), choosing Ng = 103, and in figure 3 panel (a)
+obtained for Ng = 30. Of course, the latter approach is valid as long as the low-granularity
+(low values of Ng) does not influence the qualitative properties of the kinetics.
+The second (computational) use of stochastic simulations of chemical kinetics requires a
+further discussion. At a first sight, it may appear that any stochastic simulation would be
+computationally less efficient than the solution of the corresponding mean-field equations.
+This is certainly true for classical chemical reaction schemes in a perfectly mixed system,
+7
+
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+<c>(t)
+t
+ 0
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+a
+b
+σc(t)
+t
+(a)
+(b)
+FIG. 3. Panel (a): ⟨c⟩(t) vs t at Ng = 30 (symbols) averaged over [106/Ng] realizations of the
+tank-loading process with D = 1, c0 = 1 a.u. Here, [·] indicates the integer part of its argument.
+The solid line represents the mean-field result ⟨c⟩(t) = 1 − e−t. Panel (b): Variance σc(t) vs t for
+the tank-loading process. Symbols are the results of stochastic simulations of eq. (8) averaged
+over [106/Ng] realizations, lines the solutions of eq. (10). Line (a) refers to Ng = 30, line (b) to
+Ng = 100.
+for which the mean-field model reduces to a system of ordinary differential equations for
+the concentrations of the reactants. But there are kinetic problems e.g., associated with the
+growth of microorganisms and eukaryotic cell lines in bioreactors (these growth phenom-
+ena, are indeed amenable to a description in terms of equivalent chemical reactions), the
+mean-field model of which is expressed in the form of higher-dimensional nonlinear integro-
+differential equations . For this class of problems, the use of stochastic simulations is the
+8
+
+most efficient, if not the only way to achieve a quantitative description of the process, in
+those cases where the number np of internal parameters describing the physiological state
+of an eukaryotic cell becomes large enough, np ≥ 3. This issue is addressed in detail in [27].
+This case is altogether similar to some transport problems, such as Taylor-Aris dispersion for
+high P´eclet numbers or the analysis of microfluidic separation processes (DLD devices) for
+which the stochastic simulation of particle motion is far more efficient that the corresponding
+solution of the corresponding mean-field model expressed in the form of advection-diffusion
+equations [28, 29].
+To complete the analysis of the tank-loading problem, the associated CME reads
+dp(n, t)
+dt
+= D Ng [p(n − 1, t) ηn−1 − p(n, t)] + D [(n + 1) p(n + 1, t) − n p(n, t)]
+(9)
+where ηh = 1 for h ≥ 0 and ηh = 0 otherwise. It follows that ⟨c⟩(t) = c0
+�∞
+n=1 n p(n, t)/Ng
+satisfies identically the mean-field equation (due to the linearity of the problem), while the
+variance σc(t), with σ2
+c(t) = c2
+0
+�∞
+n=1 n2 p(n, t)/N2
+g − (c0
+�∞
+n=1 n p(n, t)/Ng)2, satisfies the
+equation
+dσ2
+c
+dt = −2 D σ2
+c + D
+� 1
+Ng
++ ⟨c⟩
+Ng
+�
+(10)
+Figure 3 panel (b) compares the results of stochastic simulations against the solutions of eq.
+(10) for two values of Ng.
+The above approach can be extended to any system of nonlinear reaction schemes involv-
+ing unimolecular and bimolecular reaction, and in the presence of slow/fast kinetics. The
+structure of the reaction mechanism can be arbitrarily complicated without adding any fur-
+ther complexity (other than purely notational) in the formulation of the stochastic evolution
+expressed in terms of number of molecules. The only practical issue, is that the number
+of different families of stochastic processes grows with the number of elementary reactive
+processes considered. For instance, in the case of the subtrate-inhibited Michaelin-Menten
+kinetics
+E + S
+k1
+⇋
+k−1 ES
+ES
+k2
+→ E + P
+(11)
+ES + S
+k3
+⇋
+k−3 ESS
+there are five reactive processes (five channels in the language of the Gillespie algorithm)
+and consequently five families of counting processes {χ(h)
+ih (t, ·)}, h = 1, . . . , 5, should be
+9
+
+introduced, so that the formulation of the discrete stochastic dynamics reads
+dNS(t)
+dt
+= −
+NS(t)
+�
+i=1
+dχ(1)
+i (t, �k1 NE(t))
+dt
++
+NES(t)
+�
+j=1
+dχ(2)
+j (t, k−1)
+dt
+dNE(t)
+dt
+= −
+NS(t)
+�
+i=1
+dχ(1)
+i (t, �k1 NE(t))
+dt
++
+NES(t)
+�
+j=1
+dχ(2)
+j (t, k−1)
+dt
++
+NES(t)
+�
+h=1
+dχ(3)
+h (t, k2)
+dt
+dNES(t)
+dt
+=
+NS(t)
+�
+i=1
+dχ(1)
+i (t, �k1 NE(t))
+dt
+−
+NES(t)
+�
+j=1
+dχ(2)
+j (t, k−1)
+dt
+−
+NES(t)
+�
+h=1
+dχ(3)
+h (t, k2)
+dt
+−
+NS(t)
+�
+k=1
+dχ(4)
+k (t, �k3 NES(t))
+dt
++
+NESS(t)
+�
+l=1
+dχ(5)
+l (t, k−3)
+dt
+(12)
+dNESS(t)
+dt
+=
+NS(t)
+�
+k=1
+dχ(4)
+k (t, �k3 NES(t))
+dt
+−
+NESS(t)
+�
+l=1
+dχ(5)
+l (t, k−3)
+dt
+dNP(t)
+dt
+=
+NES(t)
+�
+h=1
+dχ(3)
+h (t, k2)
+dt
+equipped with the initial conditions cS(0) = cS,0, cE(0) = cE,0, cES(0) = cESS(0) = cP(0) =
+0. Observe that for the bimolecular steps we have used a number-dependent rate coefficient.
+This is just one possibility, out of other fully equivalent alternatives, of defining bimolecular
+reacting processes, and out of tem a numerical algorithm for solving them. This issue, and
+its computational implications will be addressed elsewhere [26]. The granularity number Ng
+can be fixed, so that
+NS(0) = [cS,0 Ng] ,
+NE,0 = [cE,0 Ng]
+(13)
+where [ξ] indicates the integer part of ξ, thus defining the relation betwen Nα(t) and cα(t),
+namely cα(t) = Nα(t)/Ng, α = S, E, ES, ESS, P. This implies also that the effective rate
+parameters entering the discrete stochastic evolution equation (12), and associated with the
+two bimolecular reactive steps, are given by �k1 = k1/Ng, and �k3 = k3/Ng.
+Consider the case k−1 = k2 = k3 = k−3 = 1, cS,0 = 4, cE,0 = 0.1. In this case the
+quasi steady-state approximation of the cES-cS curve (representing the slow manifold of the
+kinetics takes the expression
+cES =
+cE,0 cS
+KM + cS + β c2
+S
+,
+KM = k−1 + k2
+k1
+,
+β = k−3
+k3
+(14)
+Figure 4 depicts the cES-cS graph obtained from a single realization of the stochastic process
+eq. (11) at several values of k1 so as to modify the Michaelis-Menten constant KM for a
+value Ng = 106 of the granularity number.
+10
+
+ 0
+ 0.02
+ 0.04
+ 0.06
+ 0.08
+ 0
+ 1
+ 2
+ 3
+ 4
+cES
+cS
+FIG. 4. cES vs cS plot of the substrate-inhibited enzymatic kinetics discussed in the main text.
+Symbols (in color) are the results of stochastic simulations of a single realization of the process eq.
+(11), (black) solid lines the graph of the quasi steady-state approximation. The arrow indicates
+increasing values of KM, i.e. decreasing values of k1 = 20, 6, 2.
+Apart from the initial transient giving rise to an overshot in the values of cES near
+cS ≃ cS,0, the dynamics rapidly collapses towards the slow manifold and the stochastic
+simulations at high Ng-value provide a reliable description of the mean-field behavior starting
+from a single stochastic realization.
+To conclude, we want to point out some advantages and extensions of the present ap-
+proach:
+• it shows a direct analogy between chemical reaction kinetics, radiative processes and
+stochastic formulation of open quantum systems, thus, paving the way for a unified
+treatment of the interpaly between these phenomena, that is particularly important
+in the field of photochemistry, and in the foundation of statistical physics [30, 31];
+• it can be easily extended to semi-Markov transition. This is indeed the case of the
+growth kinetics of eukaryotic microorganisms, the physiological state of which can
+be parametrized with respect to internal (hidden) parameters such as the age, the
+cytoplasmatic content, etc.;
+• it can be easily extended to include transport phenomena. In point of fact, the oc-
+currence of Markovian or semi-Markovian transitions in modeling chemical kinetics is
+11
+
+analogous to the transitions occurring in the direction of motion (Poisson-Kac pro-
+cesses, L´evy flights, Extended Poisson-Kac processes) or in the velocity (linearized
+Boltzmannian schemes) [32–34].
+• it is closely related to the formulation of stochastic differential equations for the ther-
+malization of athermal system [35], in which the classical mesoscopic description of
+thermal fluctuations, using the increments of a Wiener process, is replaced by a dy-
+namic model involving the increments of a counting process.
+Due to the limitations of a Letter, all these issues will be addressed in forthcoming works. But
+apart for these extensions and improvements, the proposed formulation indicates that the
+stochastic theory of chemical reactions can be built upon a simple and consistent mathemat-
+ical formalism describing the elementary reactive events as Markovian or semi-Markovian
+counting processes [36], that perfectly fits with the description of molecular non reactive
+events (molecular collisions), providing an unifying stochastic formalism of elementary (clas-
+sical and quantum) molecular events.
+[1] P. L. Krapivsky, S. Redner, E. Ben-Naim, A Kinetic View to Statistical Physics, Cambridge
+University Press, Cambridge (2010).
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+X + A ⇋ B + X, Chemical Physics Letters, 10 (4) (1971) 371–374.
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+chemically reacting systems, The Journal of Chemical Physics 121 (2004) 4059–67.
+[20] M. Rathinam, L. R. Petzold, Y. Cao, D. T. Gillespie, Stiffness in stochastic chemically reacting
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+lation method, The Journal of Chemical Physics 124 (4) (2006) 044109.
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+[23] D. C. Venerus and H. C. ¨Ottinger, A modern Course in Transport Phenomena, Cambridge
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+(1974).
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+Royal Society of London A (235) (1956) 67-77.
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+ation (2022).
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+ford (2002).
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+nomena: Generalized Poisson–Kac processes—Part I basic theory, Journal of Physics A (50)
+(2017) 335002.
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+stochastic processes with finite propagation velocity, Physical Review X (12) (2022) 021004.
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+14
+
+matics 9 (2021) 25-73.
+15
+

4NE4T4oBgHgl3EQfbQzc/content/tmp_files/load_file.txt

@@ -0,0 +1,344 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf,len=343
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='05072v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='stat-mech]  12 Jan 2023  Chemical kinetics and stochastic differential equations  Chiara Pezzotti and Massimiliano Giona1, ∗  1Dipartimento di Ingegneria Chimica, Materiali,  Ambiente La Sapienza Universit`a di Roma  Via Eudossiana 18, 00184 Roma, Italy  (Dated: January 13, 2023)  Abstract  We propose a general stochastic formalism for describing the evolution of chemical reactions  involving a finite number of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This approach is consistent with the statistical analysis  based on the Chemical Master Equation, and provides the formal setting for the existing algorithmic  approaches (Gillespie algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Some practical advantages of this formulation are addressed, and  several examples are discussed pointing out the connection with quantum transitions (radiative  interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  ∗ corresponding author:massimiliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='giona@uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='it  1  All the chemical physical processes involve, in an atomistic perspective, a stochastic de-  scription of the events, be them reactive or associated with a change of phase (for instance  adsorption) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Nonetheless, in the overwhelming majority of the cases of practical and labo-  ratory interest, the number of molecules involved is so large to justify a mean field approach,  essentially based on the Boltzmannian hypothesis of molecular chaos (the “stosszahlansatz”)  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The mean field formulation represents the backbone of the classical theory of chemical  reaction kinetics [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  It is well known that, in all the cases where the number of molecule is small (and this  occurs in subcellular biochemical reactions, in nanoscale systems, or in the growth kinetics of  microorganisms [5–7]), the effects of fluctuations become significant, motivating a stochastic  description of chemical kinetic processes, involving the number of molecules present in the  system, thus explicitly accounting for due to their finite number [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The statistical theory  of chemical kinetics in these conditions is grounded on the Chemical Master Equation (CME)  [12, 13], expressing the evolution equation for the probabilities p(N, t) of all the possible  number-configurations N(t) = (N1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' , Ns(t)), where Nh(t) is the number of molecules of  the h-th reacting species at time t, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' However, apart from a handful of simple  cases, for which the CME can be solved analytically [14], numerical methods should be  applied to it in order to compute mean values and higher-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' But also this  choice reveals itself to be unfeasible in most of the situations of practical and theoretical  interests, due to the extremely large number of configurations involved, making the multi-  index matrix p(N, t) so huge to exceed reasonable computational facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  In order to solve this problem, Gillespie proposed an algorithmic solution to the numeri-  cal simulation of stochastic reacting systems, based on the Markovian nature of the reactive  events [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The original Gillespie algorithm has been extended and improved over time,  providing a variety of slightly different computational alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' A common denominator  of the first family of the Gillespie algorithms (namely those based on the direct method, the  first reaction method or their derivates [17–19]) is to associate to every time step the occur-  rence of just one reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This formulation comes directly from the assumption that, if the  time step is small enough, the probability that more than one reaction will occur is negligi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' While correct, this choice brings to significant computational costs for complex reaction  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This problem has been highlighted several times, from the Gillespie group itself,  as stiffness in stochastic chemical reacting systems [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' A brilliant way to overcome this  2  limit originates the famous tau-leaping method, which, unfortunately, requires to check that  the propensity functions remain almost constant at each iteration and can be applied just  if this condition is verified [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The algorithmic solution associated with the formalism  here introduced combines the accuracy of the first SSA with the computational advantages  of the τ-leaping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  There is, moreover, a missing link between the CME theory and the Gillespie algorithm,  consisting in the straight mathematical formulation of the stochastic differential equations  associated with a chemical reacting system, the statistical description of which would cor-  respond to the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' To clarify this issue, consider the conceptually analogous problem of  particle diffusion over the real line, the statistical description of which is expressed by the  parabolic equation ∂p(x, t)/∂t = D ∂2p(x, t)/∂x2, for the probability density p(x, t) of finding  a particle at position x at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Setting xn = x(n ∆t), an algorithm describing this process  can be simply expressed by the discrete evolution equation xn+1 = xn +  √  2 D ∆t rn+1, where  rh, h = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' represent independent random variables sampled from a normal distribution  (with zero mean, and unit variance) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This represents an efficient algorithmic solution  of the problem, whenever the time resolution ∆t is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Nevertheless, the mere  algorithmic approach cannot be considered physically satisfactory, in a comprehensive for-  mulation of transport theory embedded in a continuous space-time (in which both position x  and time t are real valued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' In point of fact, only with the mathematical formulation due to  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Ito of stochastic differential equations driven by the increments dw(t) of a Wiener process  (Langevin equations) [24], namely dx(t) =  √  2 D dw(t) the theory of diffusive motion has  found a proper mathematical physical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  A similar situation applies to the case of stochastic models of chemical reaction kinetics,  and the present Letter is aimed at filling this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The basic idea is that any reactive  process corresponds to a system of elementary events (the single reaction) possessing a  Markovian transitional structure, and, consequently, amenable to a description by means  of the increments of counting processes (Poisson processes, in the Markovian case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This  topic has been also pointed out in [25] in terms of Poisson measures, although the latter  formulation is much less simple and physically intuitive than the approach proposed in the  present Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  To begin with, consider the simple case of a first-order chemical reaction A  k1  ⇋  k−1 B (for  instance, an isomerization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This model is perfectly analogous to the radiative transition  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Schematic representation of the analogy between a two-level quantum system and a first-  order chemical kinetics, such as an isomerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  of a molecule possessing two energy states, due to emission and adsorption of an energy  quantum (figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Let NA(0) + NB(0) = Ng the total number of molecules at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The state of the system is characterized by the state functions σh(t), h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' , Ng for each  molecule, attaining values {0, 1}, and such that σh(t) = 0 if the energy state at time t is E0  (or equivalently if the molecule finds itself in the state A), and σh(t) = 1 in the opposite  case (energy state E1, or isomeric state B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  Let {χ(1)  h (t, k1), χ(2)  h (t, k−1)}Ng  h=1 be two systems of independent Poisson processes, char-  acterized by the transition rates k1, and k−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The evolution of σh(t) can be  expressed via the stochastic differential equation  dσh(t)  dt  = (1 − σh(t)) dχ(1)  h (t, k1)  dt  − σh(t) dχ(2)  h (t, k−1)  dt  (1)  h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' , Ng, where dχ(t, λ)/dt is the distributional derivative of the Poisson process  χ(t, λ), corresponding to a sequence of unit impulsive functions at the transition instants  t∗  i , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' , 0 < t∗  i < t∗  i+1, where for ε > 0, limε→0  � t∗  i +ε  t∗  i −ε dχ(t, λ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Summing over  h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Ng, and observing that NA(t) = �Ng  h=1 (1 − σh(t)), NB(t) = �Ng  h=1 σh(t), we have  dNB(t)  dt  =  NA(t)  �  h=1  dχ(1)  h (t, k1)  dt  −  NB(t)  �  h=1  dχ(2)  h (t, k−1)  dt  (2)  4  (b)(a)OPand dNA(t)/dt = −dNB(t)/dt, representing the evolution equation for NA(t) and NB(t),  attaining integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The stochastic evolution of the number of molecules NA(t), NB(t)  is thus expressed as a differential equation with respect to the continuous physical time  t ∈ R+, over the increments of a Poisson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Intepreted in a mean-field way, if ctot is  the overall concentration of the reactants at time t = 0, then the concentrations cα(t) at  time t can be recovered from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (2) as  cα(t) = ctot  Nα(t)  Ng  ,  α = A, B  (3)  representing the calibration relation connecting the stochastic description in terms of num-  ber of molecules Nα(t) and the concentrations cα(t), α = A, B entering the mean-field  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The analytical formulation of a stochastic differential equation for chemical kinetics,  expressed in terms of the number of molecules of the chemical species involved, rather than  an algorithm defined for discretized times, permits to develop a variety of different numerical  strategies, that naturally perform a modified tau-leaping procedure, as the occurrence of  several distinct reactive events in any elementary time step ∆t is intrinsically accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  This can be easily seen by considering the simple reaction defined by the evolution equation  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  In terms of increments, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  (2) can be written as dNB(t) = �NA(t)  h=1 dχ(1)(t, k1) −  �NB(t)  h=1 dχ(2)(t, k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' If ∆t is the chosen time step, it follows from this formulation, a simple  numerical approximation for eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (2), namely,  ∆NB(t) = NB(t + ∆t) − NB(t) =  NA(t)  �  h=1  ξ(1)  h (k1 ∆t) −  NB(t)  �  h=1  ξ(2)  h (k−1 ∆t)  (4)  where ξ(1)(k1 ∆t), ξ(2)  h (k−1 ∆t) h = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' , are two families of independent binary random  variables, where  ξ(α)  h (p) =  \uf8f1  \uf8f2  \uf8f3  1  with probability p  0  otherwise  (5)  α = 1, 2, h = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The time step ∆t, can be chosen in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (4) from the condition  K ∆t < 1 ,  K = max{k1, k−1}  (6)  In practice, we choose ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='1/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' As can be observed, the choice of ∆t is limited by the  intrinsic rates of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The advantage of deriving different algorithmic schemes for  5  solving numerically the stochastic kinetic equations becomes more evident in dealing with  bimolecular reactions (addressed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Due to the intrinsic limitations of this commu-  nication, a thorouh discussion of this issue is postponed to a future more extensive article  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The same approach can be extended to include amongst the elementary events not only  the reactive steps, but also feeding conditions, thus representing the evolution of chemically  reacting systems with a finite number of molecules in a perfectly stirred open reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This is  the case of the tank-loading problem, in which a tracer is injected in an open vessel assumed  perfectly mixed, for which, in the absence of chemical reactions, the mean field equation for  the concentration of the tracer reads  dc(t)  dt  = D (c0 − c(t))  (7)  where c0 is the inlet concentration and D the dilution rate (reciprocal of the mean retention  time), and c(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Fixing Ng so that c(t) = c0 N(t)/Ng, the corresponding stochastic  differential equation for the integer N(t) involves, also in this case, two families of counting  processes, one for the loading at constant concentration c0, and the other for tracer discharge  in the outlet stream, characterized by the same transition rate D,  dN(t)  dt  =  Ng  �  h=1  dχ(1)  h (t, D)  dt  −  N(t)  �  k=1  dχ(2)  h (t, D)  dt  (8)  starting from N(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Figure 2 depicts several realizations of the tank-loading process,  obtained by discretizing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (8) with a time step ∆t = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Despite the simplicity of the  process, this example permits to highlight the role of Ng, that can be referred to as the granu-  larity number, and the way stochastic models of chemical reactions can be fruitfully applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  Indeed, there is a two-fold use of the stochastic formulation of chemical kinetic schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The first refers to a chemical reacting system involving a small number of molecules, and  in this case Ng represents the effective number of molecules present in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The  other is to use stochastic algorithms for simulating reacting systems in an alternative (and  sometimes more efficient way) with respect to the solution of the corresponding mean-field  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' In the latter case, the granularity number Ng represents essentially a computa-  tional parameter, tuning the intensity of the fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Two choices are then possible:  (i) it can be chosen large enough, in order to obtain from a single realization of the process  an accurate approximation of the mean-field behavior, or (ii) it can be chosen small enough  6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='2  0  2  4  6  8  10  c(t)  t  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='2  0  2  4  6  8  10  c(t)  t  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='2  0  2  4  6  8  10  c(t)  t  (a)  (b)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' c(t) = N(t)/Ng vs t from a single realization of the tank-loading process eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (8) with  D = 1, c0 = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='. Panel (a): Ng = 30, panel (b) Ng = 100, panel (c) Ng = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The solid  horizontal lines represent the steady-state value c∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  in order, to deal with extremely fast simulations of a single realization of the process, that  could be averaged over a statistically significant number of realizations in due time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' These  two choices are depicted in figure 2 (panel c), choosing Ng = 103, and in figure 3 panel (a)  obtained for Ng = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Of course, the latter approach is valid as long as the low-granularity  (low values of Ng) does not influence the qualitative properties of the kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The second (computational) use of stochastic simulations of chemical kinetics requires a  further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' At a first sight, it may appear that any stochastic simulation would be  computationally less efficient than the solution of the corresponding mean-field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  This is certainly true for classical chemical reaction schemes in a perfectly mixed system,  7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='8  1  0  2  4  6  8  10  <c>(t)  t  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='2  0  2  4  6  8  10  a  b  σc(t)  t  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Panel (a): ⟨c⟩(t) vs t at Ng = 30 (symbols) averaged over [106/Ng] realizations of the  tank-loading process with D = 1, c0 = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Here, [·] indicates the integer part of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The solid line represents the mean-field result ⟨c⟩(t) = 1 − e−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Panel (b): Variance σc(t) vs t for  the tank-loading process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Symbols are the results of stochastic simulations of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (8) averaged  over [106/Ng] realizations, lines the solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Line (a) refers to Ng = 30, line (b) to  Ng = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  for which the mean-field model reduces to a system of ordinary differential equations for  the concentrations of the reactants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' But there are kinetic problems e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=', associated with the  growth of microorganisms and eukaryotic cell lines in bioreactors (these growth phenom-  ena, are indeed amenable to a description in terms of equivalent chemical reactions), the  mean-field model of which is expressed in the form of higher-dimensional nonlinear integro-  differential equations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' For this class of problems, the use of stochastic simulations is the  8  most efficient, if not the only way to achieve a quantitative description of the process, in  those cases where the number np of internal parameters describing the physiological state  of an eukaryotic cell becomes large enough, np ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This issue is addressed in detail in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  This case is altogether similar to some transport problems, such as Taylor-Aris dispersion for  high P´eclet numbers or the analysis of microfluidic separation processes (DLD devices) for  which the stochastic simulation of particle motion is far more efficient that the corresponding  solution of the corresponding mean-field model expressed in the form of advection-diffusion  equations [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  To complete the analysis of the tank-loading problem, the associated CME reads  dp(n, t)  dt  = D Ng [p(n − 1, t) ηn−1 − p(n, t)] + D [(n + 1) p(n + 1, t) − n p(n, t)]  (9)  where ηh = 1 for h ≥ 0 and ηh = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' It follows that ⟨c⟩(t) = c0  �∞  n=1 n p(n, t)/Ng  satisfies identically the mean-field equation (due to the linearity of the problem), while the  variance σc(t), with σ2  c(t) = c2  0  �∞  n=1 n2 p(n, t)/N2  g − (c0  �∞  n=1 n p(n, t)/Ng)2, satisfies the  equation  dσ2  c  dt = −2 D σ2  c + D  � 1  Ng  + ⟨c⟩  Ng  �  (10)  Figure 3 panel (b) compares the results of stochastic simulations against the solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  (10) for two values of Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  The above approach can be extended to any system of nonlinear reaction schemes involv-  ing unimolecular and bimolecular reaction, and in the presence of slow/fast kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The  structure of the reaction mechanism can be arbitrarily complicated without adding any fur-  ther complexity (other than purely notational) in the formulation of the stochastic evolution  expressed in terms of number of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The only practical issue, is that the number  of different families of stochastic processes grows with the number of elementary reactive  processes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' For instance, in the case of the subtrate-inhibited Michaelin-Menten  kinetics  E + S  k1  ⇋  k−1 ES  ES  k2  → E + P  (11)  ES + S  k3  ⇋  k−3 ESS  there are five reactive processes (five channels in the language of the Gillespie algorithm)  and consequently five families of counting processes {χ(h)  ih (t, ·)}, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' should be  9  introduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' so that the formulation of the discrete stochastic dynamics reads  dNS(t)  dt  = −  NS(t)  �  i=1  dχ(1)  i (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' �k1 NE(t))  dt  +  NES(t)  �  j=1  dχ(2)  j (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k−1)  dt  dNE(t)  dt  = −  NS(t)  �  i=1  dχ(1)  i (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' �k1 NE(t))  dt  +  NES(t)  �  j=1  dχ(2)  j (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k−1)  dt  +  NES(t)  �  h=1  dχ(3)  h (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k2)  dt  dNES(t)  dt  =  NS(t)  �  i=1  dχ(1)  i (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' �k1 NE(t))  dt  −  NES(t)  �  j=1  dχ(2)  j (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k−1)  dt  −  NES(t)  �  h=1  dχ(3)  h (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k2)  dt  −  NS(t)  �  k=1  dχ(4)  k (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' �k3 NES(t))  dt  +  NESS(t)  �  l=1  dχ(5)  l (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k−3)  dt  (12)  dNESS(t)  dt  =  NS(t)  �  k=1  dχ(4)  k (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' �k3 NES(t))  dt  −  NESS(t)  �  l=1  dχ(5)  l (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k−3)  dt  dNP(t)  dt  =  NES(t)  �  h=1  dχ(3)  h (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' k2)  dt  equipped with the initial conditions cS(0) = cS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' cE(0) = cE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' cES(0) = cESS(0) = cP(0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Observe that for the bimolecular steps we have used a number-dependent rate coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  This is just one possibility, out of other fully equivalent alternatives, of defining bimolecular  reacting processes, and out of tem a numerical algorithm for solving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This issue, and  its computational implications will be addressed elsewhere [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The granularity number Ng  can be fixed, so that  NS(0) = [cS,0 Ng] ,  NE,0 = [cE,0 Ng]  (13)  where [ξ] indicates the integer part of ξ, thus defining the relation betwen Nα(t) and cα(t),  namely cα(t) = Nα(t)/Ng, α = S, E, ES, ESS, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This implies also that the effective rate  parameters entering the discrete stochastic evolution equation (12), and associated with the  two bimolecular reactive steps, are given by �k1 = k1/Ng, and �k3 = k3/Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  Consider the case k−1 = k2 = k3 = k−3 = 1, cS,0 = 4, cE,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' In this case the  quasi steady-state approximation of the cES-cS curve (representing the slow manifold of the  kinetics takes the expression  cES =  cE,0 cS  KM + cS + β c2  S  ,  KM = k−1 + k2  k1  ,  β = k−3  k3  (14)  Figure 4 depicts the cES-cS graph obtained from a single realization of the stochastic process  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' (11) at several values of k1 so as to modify the Michaelis-Menten constant KM for a  value Ng = 106 of the granularity number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='08  0  1  2  3  4  cES  cS  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' cES vs cS plot of the substrate-inhibited enzymatic kinetics discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  Symbols (in color) are the results of stochastic simulations of a single realization of the process eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  (11), (black) solid lines the graph of the quasi steady-state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' The arrow indicates  increasing values of KM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' decreasing values of k1 = 20, 6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  Apart from the initial transient giving rise to an overshot in the values of cES near  cS ≃ cS,0, the dynamics rapidly collapses towards the slow manifold and the stochastic  simulations at high Ng-value provide a reliable description of the mean-field behavior starting  from a single stochastic realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  To conclude, we want to point out some advantages and extensions of the present ap-  proach:  it shows a direct analogy between chemical reaction kinetics, radiative processes and  stochastic formulation of open quantum systems, thus, paving the way for a unified  treatment of the interpaly between these phenomena, that is particularly important  in the field of photochemistry, and in the foundation of statistical physics [30, 31];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  it can be easily extended to semi-Markov transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' This is indeed the case of the  growth kinetics of eukaryotic microorganisms, the physiological state of which can  be parametrized with respect to internal (hidden) parameters such as the age, the  cytoplasmatic content, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  it can be easily extended to include transport phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' In point of fact, the oc-  currence of Markovian or semi-Markovian transitions in modeling chemical kinetics is  11  analogous to the transitions occurring in the direction of motion (Poisson-Kac pro-  cesses, L´evy flights, Extended Poisson-Kac processes) or in the velocity (linearized  Boltzmannian schemes) [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  it is closely related to the formulation of stochastic differential equations for the ther-  malization of athermal system [35], in which the classical mesoscopic description of  thermal fluctuations, using the increments of a Wiener process, is replaced by a dy-  namic model involving the increments of a counting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  Due to the limitations of a Letter, all these issues will be addressed in forthcoming works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' But  apart for these extensions and improvements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' the proposed formulation indicates that the  stochastic theory of chemical reactions can be built upon a simple and consistent mathemat-  ical formalism describing the elementary reactive events as Markovian or semi-Markovian  counting processes [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' that perfectly fits with the description of molecular non reactive  events (molecular collisions),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' providing an unifying stochastic formalism of elementary (clas-  sical and quantum) molecular events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Krapivsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Redner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Ben-Naim, A Kinetic View to Statistical Physics, Cambridge  University Press, Cambridge (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Boltzmann, Weitere Studien ¨uber das W¨armeglichgenicht unter Gas-molek¨ulen, Sitzungs-  berichte Akademie der Wissenschaften 66 (1872) 275-370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' Marin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfbQzc/content/2301.05072v1.pdf'}
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+MNRAS 000, 000–000 (0000)
+Preprint 1 February 2023
+Compiled using MNRAS LATEX style file v3.0
+Tracing of Magnetic field with gradients: Sub-Sonic Turbulence
+K. W. Ho, 1 ★ A. Lazarian, 1,2 †
+1Department of Astronomy, University of Wisconsin-Madison, Madison, WI, 53706, USA
+2Centro de Investigación en Astronomía, Universidad Bernardo O’Higgins, Santiago, General Gana 1760, 8370993, Chile
+Accepted 2023 January 12. Received 2023 January 12; in original form 2022 March 21
+ABSTRACT
+Recent development of the velocity gradient technique shows the capability of the technique in the way of tracing magnetic
+fields morphology in diffuse interstellar gas and molecular clouds. In this paper, we perform the numerical systemic study of the
+performance of velocity and synchrotron gradient for a wide range of magnetization in the sub-sonic environment. Addressing
+the studies of magnetic field in atomic hydrogen, we also study the formation of velocity caustics in the spectroscopic channel
+maps in the presence of the thermal broadening. We show that the velocity caustics can be recovered when applied to the Cold
+Neutral Medium (CNM) and the Gradient Technique (GT) can reliably trace magnetic fields there. Finally, we discuss the changes
+of the anisotropy of observed structure functions when we apply to the analysis the procedures developed within the framework
+of GT studies.
+Key words: ISM: structure – ISM: atoms – ISM: clouds – ISM: magnetic fields
+1 INTRODUCTION
+Magnetic fields are very important for key astrophysical processes in
+interstellar media (ISM) such as the formation of stars (see McKee
+& Ostriker 2007; Mac Low & Klessen 2004), the propagation and
+acceleration of cosmic rays (see Jokipii 1966; Yan & Lazarian 2008),
+the regulation of heat and mass transfer between different ISM phases
+(see Draine 2009 for the list of the different ISM phases). Polarized
+radiation arising from the presence of the magnetic field also inter-
+feres with the sygnal of the enigmatic CMB B-modes arising from
+gravity waves in the early Universe. (Zaldarriaga & Seljak 1997;
+Caldwell et al. 2017; Kandel et al. 2017). Therefore, it is essential to
+have a reliable way to study the properties of magnetic fields in those
+process.
+The traditional way to study the Plane of Sky (POS) magnetic fields
+is using polarimetry measurements (Planck Collaboration
+2018;
+Lazarian 2002). It is widely used from radio to optical wavelengths
+to trace the magnetic field morphology at various scales in the ISM.
+Recently, a new promising technique has been proposed, the veloc-
+ity gradient technique (VGT), which is capable of tracing magnetic
+field using spectroscopic data (Yuen & Lazarian 2017a; Lazarian et
+al. 2018; Hu et al. 2019; Ho & Lazarian 2021). The technique makes
+use of the fact that magnetic fields make turbulence anisotropic, with
+turbulent eddies being elongated along the magnetic field (See Beres-
+nyak & Lazarian (2019) for a monograph). As a result, the turbulence
+induces the fluid motion mostly perpendicular to the direction sur-
+rounding magnetic eddies. It is important that the magnetic field
+direction is the local direction of magnetic field in the vicinity of
+turbulent eddies. This follows directly from the theory of turbulent
+reconnection that predicts that magnetic fields of the eddies reconnect
+★ E-mail: [email protected]
+† E-mail: alazarian@facstaff.wisc.edu
+over one eddy turnover time (Lazarian & Vishniac (1999), hereafter
+LV99). This property of magnetic turbulence is central for magnetic
+field tracing with both velocity gradients as well as other types of
+gradients, e.g. synchrotron intensity gradients (Lazarian et al. 2017),
+synchrotron polarization gradients (Lazarian et al. 2018).
+The VGT has been numerically tested for a wide range of column
+densities from diffuse transparent gas to molecular self-absorbing
+dense gas (Yuen & Lazarian 2017a; Lazarian & Yuen 2018a; Hu
+et al. 2019; Hu & Lazarian 2021). The technique was shown to
+be able to provide both the orientations of the magnetic field as
+well as a measure of media magnetization (Lazarian et al. 2018).
+A VGT survey was conducted recently to study the morphology of
+a few nearby molecular cloud (Hu et al. 2019). The result showed
+consistency with the Planck polarization measurement and indicate
+the capability of the VGT on tracing magnetic field in different ISM
+region.
+While the earlier VGT study mainly focused on the supersonic
+spectroscopic data, the same idea of tracing magnetic with gradients
+can be employed with different types of astrophysical data. For in-
+stance, Lazarian et al. (2017) showed gradient can also be applied
+to trace magnetic field with synchrotron intensity gradients (SIGs)
+maps. The corresponding emission comes from subsonic warm/hot
+media. The lack of shock wave in sub sonic environment is beneficial
+for magnetic field tracing.
+Tracing of magnetic field in subsonic media is also important
+within the VGT. The velocity gradients can be obtained in this setting
+using velocity centroids which are not sensitive to thermal broaden-
+ing. If the channel maps are applied to subsonic data, first of all, one
+can use heavier species as spectroscopic tracers. For such species,
+the thermal broadening is suppressed and caustics produced in chan-
+nel maps are prominent. In addition, the newly introduced Velocity
+Decomposition Algorithm (VDA) Yuen et al. (2021) opens ways of
+exploring velocity caustics in the presence of the thermal broadening.
+© 0000 The Authors
+arXiv:2301.13458v1  [astro-ph.GA]  31 Jan 2023
+
+2
+Ho & Lazarian
+Therefore this study explores the ability of magnetic field tracing
+using both the VGT and the SIGs for subsonic medium. Several
+concerns arise on the application of Gradient Technique (GT) in the
+sub-sonic environment. First, multi-phase media study (see Yuen et
+al. (2021)) shows that thermal broadening is a crucial factor that
+smooths out the structure in the subsonic spectroscopic data. It may
+potentially weaken the ability of the VGT to trace the magnetic field.
+Second, Ho & Lazarian (2021) found out that the intermittency of fast
+mode could also play an important role in affecting the VGT analysis.
+In the case that the fast mode dominate the energetics of a particular
+region they induce there the rotation of the velocity gradient direction
+from parallel to perpendicular to the magnetic field. This, however,
+does not happen with the SIGs, for which the gradients induced by
+fast and Alfven modes are parallel.
+In Ho & Lazarian (2021) we proposed a new technique, Gradient
+of Gradient Amplitude (GGA), which improves the magnetic field
+tracing by gradients. However, an in-depth study is required to ana-
+lyze the applicability of GGA in sub-sonic regime versus the change
+of Alfven Mach number.
+Below, we perform a new study of the GT in the sub-sonic environ-
+ment to answer the concerns above and evaluated the performance
+of the GT in a low Ms regime. In what follows, we would cover the
+theory in section 2 and our numerical setup in section 3. Then we
+would discuss the result of the alignment measure of the gradient in
+the ideal observable measure and the velocity gradient in the pres-
+ence of thermal broadening in multi-phase media in section 4. We
+further extend the study of GGA in section 5. We then discuss the the
+Correlation Function Analysis (Hereafter CFA) alignment in section
+6. At last, we would discuss our work in section 7 and summarize the
+paper in section 8.
+2 GRADIENT TECHNIQUE
+2.1 Theoretical Considerations
+The most important component of Magnetohydrodynamic (MHD)
+turbulence is the cascade of Alfvenic motions. Therefore, below we
+will focus on the properties of Alfven modes.
+The modern theory of MHD turbulence originates from the work
+of Goldreich & Sridhar 1995 (henceforth GS95) that described the
+scaling of transAlfvenic incompressible turbulence in what is now
+known to be the strong MHD turbulence regime. The description
+was, however in the frame of the mean magnetic field, which, as it
+was shown by the later studies, the GS95 statitical scalings are not
+applicable.
+Further advances were related to understanding of the importance
+of the local system of reference as well as the generalization of the
+theory for the sub-Alfvenic regime in Lazarian & Vishniac 1999
+(henceforth LV99). There also the regime of weak turbulence was
+quantified (see also Galtier et al. (2000)).
+The local system of reference is the system of reference in respect
+to which the turbulent motions should be considered. Its importance
+is easiest to see considering magnetic eddies. Due to fast turbulent
+reconnection the eddies aligned with the magnetic field direction
+in their vicinity can reconnect and perform a turnover within one
+eddy turnover time (LV99). This happens on the eddy turnover scale
+∼ 𝑙⊥/𝑣𝑙,
+where 𝑙⊥, 𝑣𝑙 are the size of eddy perpendicular to the
+local magnetic field direction and the eddy’s velocity at the scale l.
+Incidentally, this mixing results in inducing an Alfven perturbation
+with the same period, i.e. 𝑙⊥/𝑣𝑙 ∼ 𝑙∥/𝑉𝐴, where 𝑉𝐴 is the Alfven
+velocity. The latter corresponds to the condition termed "critical
+balance" in GS95. However, unlike the origianal GS95 claim, the
+critical balance is only in the system of reference aligned with the
+local direction of the magnetic field, i.e. with the direction of the
+magnetic field in the direct vicinity of the eddy. The local system of
+reference is absolutely critical for the GT. It is only because of the
+localized alignment that the gradients of velocity and magnetic field
+can trace 3D magnetic field.
+The numerical study in Cho & Vishniac 2000; Maron & Goldreich
+2000 established numerically the vital importance of the local system
+of reference for the description of MHD turbulence. The subsequent
+studies in Lithwick & Goldreich (2001) as well as Cho & Lazarian
+(2002, 2003); Kowal et al. (2009), extended the the theory to the
+compressible case. This theory of MHD compressible turbulence
+(see the monograph by Beresnyak & Lazarian (2019)) is at the basis
+of the GT.
+It is important to note that the motions perpendicular to the lo-
+cal magnetic field have the form of Alfvenic eddies and they ex-
+hibit Komlogorov scaling 𝑣𝑙 ∼ 𝑙1/3
+⊥ . Therefore the gradients scale
+as 𝑣𝑙/𝑙⊥ ∼ 𝑙−2/3
+⊥
+, meaning that the gradients at the smallest re-
+solved scales are the most important (see Lazarian et al. (2020) for
+the analytical theory of gradient measurements). These gradients are
+perpendicular to the magnetic field and their direction should be
+turned 90 degrees to get the magnetic field tracing. It is important
+that the amplitude of the gradients increases with the decrease of the
+scale. Therefore, the gradients measured at the smallest scales are
+the most prominent. These gradients, similar to aligned grains (see
+(Andersson et al. 2015)), sample the 3D magnetic field along the
+line of sight. Due to this effect, the large scale gradients, e.g. arising
+from galactic shear, are not important for the analysis of the high
+resolution data.
+2.2 Velocity and magnetic gradients
+2.2.1 General outlook
+The 3D velocity fluctuation are not directly available from the obser-
+vations. Instead, the gradients of velocity centroids and the gradients
+of intensity fluctuations measured within thin channel maps 1 can be
+used as proxies of the velocity gradients. In both cases, the gradients
+are measured for turbulent volume extended by L > 𝐿𝑖𝑛 𝑗 along the
+LOS, and this entails additional complications, where L, 𝐿𝑖𝑛 𝑗is the
+LOS depth and the injection scale. While eddies stay aligned with
+respect to the local magnetic field, the direction of the local magnetic
+field is expected to change along the LOS. Thus, the contribution of
+3D velocity gradient are also summed up along the line of sight.
+The spectrum of observed fluctuations changes due to the averag-
+ing effect along the LOS. It is easy to show that the 2D spectrum
+of the turbulence obtained by projecting the fluctuations from 3D
+has the same spectral index of -11/3 2. The relation between the
+spectral slope of the correlation function and the slope of the turbu-
+lence power spectrum in 2D in this situation is −11/3 + 2 = −5/3,
+where 2 is the dimensionality of the space. Therefore, the 2D velocity
+fluctuations arise from the 3D Kolmogorove-type turbulence scale
+as 𝑙5/6
+2𝐷 with the gradient anisotropy scaling as 𝑙−1/3
+2𝐷 . It is important
+that the amplitude of the gradients increases with the decrease of the
+1
+For a channel maps with channel width Δ𝑣, the thin channel map means
+its Δ𝑣 ≤
+√︃
+𝛿𝑣2
+𝑅, where 𝛿𝑣𝑅 is the velocity dispersion.
+2 Starting from 1D spectrum 𝑃1𝐷 with spectral index -5/3, we can get back
+3D spectral index of −11/3 by considering the dimensional analysis of 𝑃3𝐷
+= 𝑃1𝐷𝑘−2
+MNRAS 000, 000–000 (0000)
+
+3
+scale. Therefore, the gradients measured at the smallest scales are
+the most prominent. These gradients, similar to aligned grains (see
+(Andersson et al. 2015)), sample the 3D magnetic field along the
+line of sight. Due to this effect, the large scale gradients, e.g. arising
+from galactic shear, are not important for the analysis of the high
+resolution data.
+The slow modes follow the scaling of the Alfven modes (Goldreich
+& Sridhar 1995; Lithwick & Goldreich 2001; Cho & Lazarian 2002,
+2003) and therefore induce the same type of gradients as Alfvenic
+modes. while fast modes are different (Cho & Lazarian 2002, 2003;
+Kowal et al. 2009; Ho & Lazarian 2021)). It follows from the the-
+ory in (Lazarian & Pogosyan (2012), hereafter LP12) that gradients
+of synchrotron emission arising from fast modes are also aligned
+perpendicular magnetic field direction, while the anisotropies of the
+gradients of velocity caustics and velocity centroids are different
+(Kandel et al. 2017, 2018). It is possible to show (Lazarian et al.
+2018) that the corresponding gradients are perpendicular to those
+created by Alfven and slow modes. Therefore, the contribution of the
+fast modes can decrease the accuracy of the GT. We are dealing with
+their contribution in this paper.
+2.2.2 VGT for molecular clouds and diffuse HI
+The magnetic field tracing with velocity gradients in molecular
+clouds can be tested successfully with isothermal numerical sim-
+ulations (see Hu et al. (2019)). This is due to efficient cooling of
+the molecular clouds, which is different from HI gas (See Field et
+al. (1969); Wolfire et al. (1995, 2003)). The HI gas is stabilized by
+the thermal equilibrium between the heating and cooling and forms
+two stable phases: the warm and cold phases. Other than the two
+phases, the thermally unstable phase also plays a vital role in the
+atomic hydrogen environment due to the consequence of strong tur-
+bulence. Due to the presence of magnetized turbulence in the atomic
+hydrogen it is a promising medium of applying the VGT. In such an
+environment, the VGT has already demonstrated the reliable tracing
+of the magnetic field (Yuen & Lazarian 2017a; Hu et al. 2019).
+The turbulence is subsonic in most volume of galactic HI, which
+corresponds to the warm phase.(Saury et al. 2014; Marchal, Mar-
+tin & Gong 2021) The Velocity Decomposition Algorithm (VDA)
+developed in Yuen et al. (2021) allows to identify velocity caustics
+produced in this phase.
+2.3 Velocity caustics
+The concept of velocity caustics is first proposed by Lazarian &
+Pogosyan (2000) and further facilitated by Yuen et al. (2021). Veloc-
+ity caustics describes the effect of pure turbulent velocity fluctuation
+and how they come into the thin channel map. One ideal picture would
+be, even though considering a incompressible magnetized turbulent
+fluid with no density fluctuation, we can still observe a channel map
+with anisotropic fluctuation arising from the turbulence. Those fluc-
+tuations are often referred to as the velocity contribution and different
+statistical tools (for example, VGT) could utilize the information to
+trace magnetic field. However, the fluid contains compressibility and
+density contamination caused by thermal broadening effect, making
+the fluctuation of channel map contains the contribution from both
+density and velocity part. Nonetheless, the density effect on sub-sonic
+media is sub-dominate and can be removed by using the algorithm
+proposed by Yuen et al. (2021).
+2.3.1 Synchrotron emission
+Measurements of polarized synchrotron radiation and Faraday ro-
+tation (see Beck & Wielebinski (2013); Oppermann et al. (2015);
+Fletcher et al. (2011); Lenc et al. (2016); Van Eck et al. (2017) )
+provide an important insight into the magnetic structure of the Milky
+Way and the neighboring galaxies. Synchrotron radiation fluctuation
+carries the statistical information of MHD turbulence. Serial studies
+discussed how to apply gradient onto measurable quantities, such as
+synchrotron intensity and synchrotron polarization (See Lazarian et
+al. (2017); Lazarian & Yuen (2018a)). In this paper we focus on the
+gradient on synchrotron intensity map as it is a observable that we
+deal with.
+For the power-law distribution of electrons 𝑁(𝐸)𝐸 ∼ 𝐸 𝛼𝑑𝐸, the
+synchrotron emissivity is
+𝐼𝑠𝑦𝑛𝑐(X) ∝
+∫
+𝑑𝑧𝐵𝛾
+𝑃𝑂𝑆(X, z)
+(1)
+where 𝐵𝛾
+𝑃𝑂𝑆 =
+√︃
+𝐵2𝑥 + 𝐵2𝑦 corresponds to the magnetic field com-
+ponent perpendicular to the line of sight, X is the plane of sky vector
+defined in x and y direction, z the line of sight axis and, 𝐵𝑥, 𝐵𝑦 the
+3D magnetic field in x and y direction. The fractional power of the
+index 𝛾 = (𝛼 + 1)/2 was a impediment for quantitative synchrotorn
+statistical studies. However, LP12 showed that the correlation func-
+tions and spectra of the 𝐵𝛾
+⊥ could express as 𝛼 = 3, which gives 𝛾
+and therefore the dependence of synchrotron intensity on the squared
+magnetic field strength.
+2.4 Application of Gradient in Sub-Sonic Environment
+Below we will discuss two important examples to which we will apply
+the GT. Those are the centroid map and the synchrotron intensity
+map. We will perform a systematic study of the GT by changing
+the magnetization of the numerical data used to produce synthetic
+observations. Other than that, we would also like to study the behavior
+of GT in the HI spectroscopic velocity channel maps due to the recent
+debate of the velocity caustics effect in the channel map (See section
+4.3 and 7.2 for more information).
+3 NUMERICAL SIMULATION AND MEASURES
+EMPLOYED
+3.1 Simulation Setup
+The numerical data that we analyzed in this work are obtained by 3D
+MHD simulations using the single-fluid, operator-split, staggered-
+grid MHD Eulerian code ZEUS-MP/ 3D (Hayes et al. 2006) to set up
+a 3D, uniform, and isothermal turbulent medium. Periodic boundary
+conditions are applied to emulate a part of the interstellar cloud.
+Solenoidal turbulence injections are employed. To extend our study
+from super sonic regime to sub sonic regime, we simulate two sets
+of ensemble in each regime. Two sets of simulations employ various
+Alfvenic Mach numbers 𝑀𝐴 = 𝑉𝐿/𝑉𝐴 with Sonic Mach Number
+𝑀𝑆 = 𝑉𝐿/𝑉𝑆 at about 6 and 0.5 where 𝑉𝐿 represents the injection
+velocity, 𝑉𝐴 the Alfven velocities, 𝑉𝑠 the sonic velocity. For the
+generation of turbulence, the turbulence is injected solenoidally for
+all the simulations using the Fourier-space method. Turbulent energy
+is injected at the large scale ( k=2 ) and dissipated by the viscosity
+at small scale. We adjust the strength of the injection such that the
+cubes reach desired 𝑀𝑠 value. All of the cubes are listed in Table
+1. However, limited by the turbulence scaling (Please see LV99), we
+MNRAS 000, 000–000 (0000)
+
+4
+Ho & Lazarian
+Subsonic
+Supersonic
+Model
+𝑀𝑆
+𝑀𝐴
+Model
+𝑀𝑆
+𝑀𝐴
+H1S
+0.67
+0.13
+H1
+7.31
+0.22
+H2S
+0.64
+0.38
+H2
+6.10
+0.42
+H3S
+0.62
+0.64
+H3
+6.47
+0.61
+H4S
+0.61
+0.90
+H4
+6.14
+0.82
+H5S
+0.61
+1.17
+H5
+6.03
+1.01
+H6
+6.02
+1.21
+Table 1. Simulation parameters where 𝑀𝑆, 𝑀𝐴 represents the sonic Mach
+number and Alfvenic Mach number. For all simulations, the resolution is set
+to 7923. 𝑀𝑆, 𝑀𝐴 are the sonic Mach number and the Alfvenic Mach number.
+devote most of our research to the sub-Alfvenic and trans-Alfvenic
+case in this study.
+3.2 Plane of sky magnetic field
+We trace the plane of sky (POS) magnetic field orientation with
+polarization. We shall assume a constant-emissivity dust grain align-
+ment process. As a comparison to gradient, we generate polarization
+maps by projecting our data cubes along the z-axis. We construct an
+synthetic Stokes parameters Q, U.
+By assuming that the constant emissivity and the dust followed the
+gas, which the dust uniformly aligned with respect to the magnetic
+field, the Stokes parameter 𝑄(X),𝑈(X) can than be expressed as a
+function of angle 𝜃 at plane of sky magnetic field by tan(𝑥, 𝑦) =
+𝐵𝑦(𝑥, 𝑦)/𝐵𝑥(𝑥, 𝑦) :
+𝑄(X, 𝑧) ∝
+∫
+𝑑𝑧 𝜌(X, 𝑧)𝑐𝑜𝑠(2𝜃(X, 𝑧))
+𝑈(X, 𝑧) ∝
+∫
+𝑑𝑧 𝜌(X, 𝑧)𝑠𝑖𝑛(2𝜃(X, 𝑧)),
+(2)
+where 𝜌 is the density, X is the plane of sky vector defined in x and y
+direction, z the line of sight axis and, 𝐵𝑥, 𝐵𝑦 the 3D magnetic field
+in x and y direction. The dust polarized intensity 𝐼𝑃 =
+√︁
+𝑄2 + 𝑈2and
+angle 𝜃 𝑝 = 0.5𝑎𝑡𝑎𝑛2(𝑈/𝑄) are then defined correspondingly.
+3.3 Synchrotron intensity map
+For our present paper, we follow the approach in LP12 that amplitudes
+of Stokes parameters are scaled up with respect to the cosmic-ray
+index and the spatial variations of the Stokes parameters are similar
+to the case of cosmic-ray index 𝛾 = 2 .
+3.4 Alignment Measure (AM) and sub-block averaging
+To quantify how good two vector fields are aligned, we employ
+the alignment measure that is introduced in analogy with the grain
+alignment studies (see Lazarian 2002):
+𝐴𝑀 = 2⟨cos2 𝜃𝑟⟩ − 1,
+(3)
+as discussed for the VGT in González-Casanova & Lazarian 2017;
+Yuen & Lazarian 2017a). The range of AM is [−1, 1] measuring
+the relative alignment between the 90𝑜-rotated gradients and mag-
+netic fields, where 𝜃𝑟 is the relative angle between the two vectors.
+A perfect alignment gives 𝐴𝑀 = 1, whereas random orientations
+generate 𝐴𝑀 = 0 and a perfect perpendicular alignment case refers
+to 𝐴𝑀 = −1 . In what follows we use 𝐴𝑀 to quantify the alignments
+of VGT in respect to magnetic field.
+We adopt the sub-block averaging introduced in Yuen & Lazarian
+(2017a). The use of sub-block averaging comes from the fact that
+the orientation of turbulent eddies with respect to the local magnetic
+field is a statistical concept. In real space the individual gradient
+vectors are not necessarily required to have any relation to the local
+magnetic field direction. Yuen & Lazarian (2017a) reported that the
+velocity gradient orientations in a sub-region–or sub-block–would
+form a Gaussian distribution in which the peak of the Gaussian fit
+reflects the statistical most probable magnetic field orientation in this
+sub–block. As the area of the sampled region increases, the precision
+of the magnetic field traced through the use of Gaussian block fit
+becomes more and more accurate. We will discuss it more in section
+5.
+4 RESULTS
+For observational tracing of the magnetic field, it is essential to know
+what to expect in terms of AM dependence on magnetization when
+we employ the gradient method in the ideal synthetic environment.
+We investigate how the change in Alfvenic Mach number 𝑀𝐴 would
+alter the tracing power of Gradient Technique (GT) with two types
+of data: spectroscopic maps and synchrotron intensity map.
+4.1 Gradients of Synchrotron Intensity
+The synchrotron intensity gradient (SIG) results are presented in the
+left panel of Figure 1. We adopt the sub-block averaging approach,
+and the results are computed using the block size of 722. To compare
+the change of tracing power of GT in different hydro-dynamical
+regimes, we include the result of supersonic simulation(𝑀𝑆 ∼ 6)
+with similar coverage of 𝑀𝐴 as a reference. The setting of block size
+is the same as the sub-sonic regime.
+Throughout the change of 𝑀𝐴, the tracing power of SIG shows a
+different trend in different hydro-dynamical regimes. The result of
+sub-sonic environments (Blue curve) shows that the tracing power of
+SIG is insensitive to the change of magnetization. The AM maintains
+at about 0.8 with a mild drop in 𝑀𝐴 ∼ 0.4 case. For the case
+of supersonic, we observe a steady downtrend of 𝐴𝑀 in the sub-
+Alfvenic regime. The 𝐴𝑀 starts at ∼ 0.58 at 𝑀𝐴 ∼ 0.2 and drops
+gradually to 0.38 at 𝑀𝐴 ∼ 0.8. The declining trend disappears at
+the trans-Alfvenic and super-Alfvenic regime, which the AM steady
+at around 0.38. Besides, we notice that the AM of SIG in sub-sonic
+ensembles always higher than supersonic ensembles.
+4.2 Result of Gradient in Centroid
+For the benchmark of Velocity centroid gradient (VCG) in the sub-
+sonic environment, Figure. 1 showed the change of AM of centroid
+as a function of 𝑀𝐴 in the right panel. The sub-block setting is the
+same as SIG. As a reference, we also add the change of AM for the
+supersonic environment in orange color. We observe that the AM of
+VGT behaves as a monotonic function of 𝑀𝐴 in the sub-Alfvenic
+regime for both hydro-dynamical regimes. The 𝐴𝑀 declines when
+𝑀𝐴 increased. The 𝐴𝑀 continues the declining trend throughout
+from sub-Alfvenic to trans-Alfvenic regimes. However, similar to the
+SIG result for supersonic ensembles, the 𝐴𝑀 of VGT for supersonic
+ensembles becomes stable at about 0.4 at the transition from trans-
+Alfvenic to the super-Alfvenic regime,
+A tendency of well alignment between VGT and magnetic field in
+the sub-sonic case is observed here. The AM of sub-sonic set always
+better than supersonic case with the AM improvement of about 0.2
+throughout the change of 𝑀𝐴 from 0.2 to 1.2.
+MNRAS 000, 000–000 (0000)
+
+5
+Figure 1. Left panel: Result of Synchrotron intensity gradient . Right panel: Result of Centroid Gradient. Both block size used = 72. X-axis: Alfvenic Mach
+Number 𝑀𝐴, y-axis: AM. The blue lines represent the AM of sub-sonic ensembles and orange lines represent the change of AM of super-sonic ensembles.
+Figure 2. The Comparison of intensity structure under the influence of thermal broadening. Simulation used in the figure: H4S. Warmer color means denser
+pixels and coolers means pixels with lower density. The blue and red arrow represents the magnetic field direction and gradient direction within the sub-block
+(block size = 662). The bottom right shows the alignment measure value between magnetic field and gradient for each maps.
+4.3 Velocity Channel gradient in the multi-phase Interstellar
+medium
+The Velocity Channel Gradients provide another way to study the
+magnetic field’s morphology in the interstellar medium Lazarian &
+Yuen (2018a). The intensity fluctuation is strongly affected by its
+width and the thermal properties of the medium. Hu et al. (2019)
+demonstrated the reliable performance of VChGs in tracing the mag-
+netic field directions in super-sonic molecular clouds. However, con-
+cerns of the thermal broadening effect were raised in a sub-sonic
+environment, which the effect could smooth out the velocity caustics
+in the channel maps (Clark et al. 2019). In the extreme case, when
+the thermal width larger than the velocity dispersion width, the fine
+structure of the channel map would be washed out. In addition, this
+can makes it similar to the intensity map. However, the physics of
+the interstellar medium is complicated and involves external physical
+processes, especially for the HI medium. Thermal instability plays a
+crucial role in shaping the proprieties of the HI medium, resulting
+in the multi-phase interstellar medium. In multi-phase media, the
+numerical study found that the warm phase gas occupies most of
+the medium with about 5000K. On the other hand, the cold phase
+medium cools down to about 100K and occupied about 10% space (
+Heiles & Troland 2003; Kritsuk et al. 2017; Ho , Yuen & Lazarian
+2022).
+Since two-phase media has a dramatic difference in temperature,
+MNRAS 000, 000–000 (0000)
+
+Synchrotron Intensity
+Centroid
+1.0
+Sub-Sonic
+Super-Sonic
+0.8
+0.6-
+AM
+0.4 -
+0.2 
+0.0 -
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+MA
+MABroadening with warm gas only
+No Broadening
+Broadening with cold & warm gas
+M
+= 0.68
+AM = 0.96
+0.94
+Broadening like
+Velocity castics like6
+Ho & Lazarian
+the influence of broadening effect on the intensity structure in the
+channel map behaves entirely differently. The velocity profile of
+warm phase gas will greatly be extended because of its tempera-
+ture and its fine structure in the channel map being affected. As a
+result, when we look at the transition of fine structure in channel
+map when switching different velocity channel, the caustics created
+in channel maps by turbulence in the warm phase gas will lose their
+contrast due to thermal broadening. A new technique, namely, the
+Velocity Decomposition Technique (VDA) can deal with the effect
+of thermal broadening and focus on the velocity caustics (Yuen et
+al. 2021). In what follows, we another way of how the dynamics of
+warm gas can be revealed in the multi-phase medium.
+If the multi-phase media is a unified turbulent system, dynamics
+between cold gas and warm gas are coupled (Yuen et al. 2022).
+The cold phase gas forms clumps that moving with the surrounding
+warm gas. It suggests the dynamical information of warm phase gas
+will imprint in the cold phase that is not much affected by thermal
+broadening. We expect this effect to be important in multi-phase
+galactic HI.
+To explore and verify this effect, we adopt a post-processing analy-
+sis to make synthetic observation of a multi-phase environment with
+broadening based on our sub-sonic ensembles simulation set. In our
+synthetic observation , we randomly select 15% of pixels and label
+them as a cold phase gas tracer. We label the rest of the pixels as
+warm phase gas. We then transform the Position Position Position
+data cube (PPP) to Position Position Velocity (PPV) cube. We cal-
+culate a PPV cube accounting for a broadening effect. To do so, we
+convolved each pixel with its temperature profile. To simplify our set
+up, we set the temperature of warm gas as 5000K and 100K for cold
+gas. The idea of the post-processing synthetic observation is inspired
+by Yuen et al. (2021). As noticed in Lazarian & Pogosyan (2000), the
+fluctuation of channel maps can be divided into those arising from
+density and velocity. It is demonstrated in Yuen et al. (2021) that,
+without changing the density value, one can vary the sound speed to
+change the fraction of density and velocity contributions in a channel
+map. We should stress that the isothermal simulation could not cap-
+ture the full physics in multiphase ISM. However, Yuen et al. (2021)
+demonstrated that the contribution of CNM and WNM in channel
+map could also be separated into the density and velocity part with
+the difference of different thermal profile. As a result, we can apply
+two thermal profiles to the gas to try to simulate the behaviour of
+CNM and WNM in a channel map.
+Figure 2 demonstrates the center channel Map of synthetic obser-
+vation from one of our simulation cubes(Right). As a reference, the
+figure also includes two comparison plots of the same Channel Map
+but one with a broadening effect with only warm phase (Left) and
+another one without broadening(Mid). This two picture represents
+two different regimes. In the sub-sonic regime, the morphology of
+the channel map without broadening shows a reference of intensity
+fluctuations caused by velocity caustics. Because of the existence of
+the velocity caustics effect, the channel map structure without broad-
+ening effect would demonstrate an intensity structure, which filling
+with thin and long filaments. Those intensity filaments caused by
+caustics within the thin channel map are elongated along the mag-
+netic field, as described in LY18. On the contrary, the morphology of
+the channel map dominated by the broadening effect is different. In
+particular, the intensity fluctuation in the channel map is washed out
+because of the wide thermal velocity profiles. Therefore, the intensity
+structure in the channel map has a high similarity with the intensity
+maps. The similarity of the effects of thermal broadening and the
+increase of the thickness of the channel maps is discussed in LP00.
+The situation is changed if we observe the intensity of emission in
+Figure 3. Result of Channel Gradient considering the effect of thermal broad-
+ening. Block size used = 66. X-axis: Alfvenic Mach Number 𝑀𝐴, y-axis: AM
+thin channel maps arising from the mixture of warm and cold gas.
+There, the thin and long filamentary structures are clearly seen. This
+suggests that the main structure of velocity caustics is preserved in
+the the presence of multi-phase media with cold and warm gas mixed
+together.
+Figure 3 shows a scatter and line plot of AM of VGChT using
+channel map of multi-phase synthetic simulation with respect to 𝑀𝐴
+using the gradient recipe same as the Figure 1. The plot includes the
+𝐴𝑀 obtained in the channel maps with and without broadening. The
+AM for multi-phase simulation starting with 𝐴𝑀 ∼ 1.0 in 𝑀𝐴 ∼ 0.2
+with slowly decline to 𝐴𝑀 ∼ 0.88 in 𝑀𝐴 ∼ 1.2. In contrast to the
+broadening regime, the AM curve for multi-phase simulation is very
+close to the velocity caustics regime in the sub-Alfvenic simulation
+with a small difference of AM. This discrepancy becomes broader
+as we transfer to the trans-Alfvenic environment.
+5 IMPROVING AM IN SUB-SONIC MAP USING GGA
+TECHNIQUE
+Ho & Lazarian (2021) identified the effect of intermittency of fast
+mode in low-plasma 𝛽 media. Therefore, the concentration of fast
+modes in selected regions would alter the anisotropy of the distri-
+bution of velocity centroids compared to the neighboring regions
+dominated by Alfvenic modes (see Kandel et al. (2018)).
+This effect would be reflected in the observed centroid gradients
+to abruptly change 90 degrees in the fast mode dominated regions.
+We refer those gradients as orthogonal gradient. Ho & Lazarian
+(2021) introduced new data sets, namely, gradient amplitude map,
+and demonstrated that using these data sets one could suppress the
+orthogonal gradient effect. As a result, the new gradient technique,
+Gradient of Gradient Amplitudes (hereafter GGA), could improve
+the alignment measure. The performance of GGA in ideal case (En-
+MNRAS 000, 000–000 (0000)
+
+Channel map with the thermal broadening effect
+1.00
+0.95
+0.90
+0.85
+AM
+0.80
+0.75
+No broadening
+Cold gas included
+0.70
+Warm gas only
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+MA7
+Figure 4. The AM of GGA versus the block size using synchrotron intensity.
+The line with different colors represent the performance of GGA with certain
+strength of white noise added. As a reference, the dotted line with red color
+illustrate the performance of gradient with the noise amplitude of 1𝜎. The
+x-axis showed in log scale for demonstrating the performance of technique in
+small block size.
+Simulation used: H1S
+Block size covered: [11,18,22,33,36,44,66,72,99,132,198,396]
+vironment without noise) could provide prefect alignment (AM∼ 1)
+with the use of block size larger than 502.
+However, we noticed that the performance of GGA could strongly
+depends on the level of noise. The performance of GGA will declin
+rapidly with the increase of noise. To demonstrate the effect of GGA
+in the presence of noise, we add white noise with the amplitude
+relative to the standard deviation of the observable measures and see
+how the 𝐴𝑀 of GGA is varied as a function of noise amplitude.
+Figure 4 shows the 𝐴𝑀 of GGA in centoid maps versus block size
+with white noise added of the amplitude 0.05 𝜎 and 0.1 𝜎. As a
+reference, we also added the AM of GGA without noise. Also, we
+include the AM of gradient with noise of 0.1 𝜎 for a comparison.
+For the computation of GGA, we first define the gradient amplitude
+map (GA), which mechanistically defined as 𝐺𝐴 = �
+𝑖 𝐴2
+𝑖 , where 𝐴𝑖
+is gradient component in direction i. For the gradient technique, 𝐴𝑖
+can be computed though the Sobel kernel. The GGA would then be
+the output of the Sobel kernel of GA.
+One can see from the figure, the performance of GGA drops rapidly
+with mild noise added. Compare to ideal case, the AM of GGA falls
+from ∼ 0.9 to ∼ 0.6 in the small block size For noise amplitude of
+0.05𝜎. The performance gap narrows down with the larger block size
+but block size of ≥ 1202 is required to match the performance of ideal
+case. The advantage of GGA over ordinary gradient decreases for the
+case of noise amplitude 0.1 𝜎. We can see that the performance of
+GGA is very sensitive to the noise level if we use a smaller block
+size.
+To restore the performance of GGA, we employ the Gaussian
+smoothing of 𝜎 = 2 pixel as proposed in Lazarian et al. (2017) and
+tested in Lazarian & Yuen (2018a). According to Lazarian et al.
+Figure 5. The comparison of GGA before and after the smoothing technique
+using the synchrotron intensity map. As a reference, a blue line is added for
+representing the idea case.
+Simulation used: H1S
+Block size covered: [11,18,22,33,36,44,66,72,99,132,198,396]
+(2017), the kernel size we picked here would preserve most of the
+small-scale structures while efficiently suppressing the noise in the
+synthetic map globally. By adding the noise and also the smoothing
+kernel, we can then test whether in noisy observations we can still
+use the GGA as a tool to trace magnetic field. Figure 5 shows the
+result of GGA verus block size with noise added of amplitude 0.1𝜎
+and smoothing. The setup is the same as Figure 4. We can see that the
+application of the smoothing technique shows that the performance of
+GGA can be improved. The drop of AM from 0.5 decrease to 0.8 in the
+small block size while the performance gap between smoothing and
+ideal case become negligible in the block size of 602. The smoothing
+technique could relax the noise level requirement of the GGA.
+6 CFA IN GRADIENT AMPLITUDE MAP
+Other than gradient, Correlation Function Analysis(CFA) is another
+technique of tracing magnetic field direction by utilizing observable
+measure information (Esquivel & Lazarian 2005; Kandel et al. 2017;
+Hernández-Padilla et al. 2020). CFA was suggested to study magnetic
+field statistically and it is based on the theoretical understanding of
+properties of observed fluctuations (see LP12). For the (2 order)
+correlation function 𝐶𝐹𝐶 of a velocity centroid map 𝐶, it is defined
+as
+𝐶𝐹𝐶 (R) =< 𝐶(r)𝐶(r + R) >,
+(4)
+where 𝑟, 𝑅 are the vector quantities on 2D maps and separation
+distance from r. The output of 2D correlation map 𝐶𝐹𝐶 (R) can
+be interpreted as the fluctuations between different distance R. If
+the fluctuations are isotropic, the shape of contour line will be cir-
+cular. In opposite, the shape turns to elliptical when the fluctuation
+MNRAS 000, 000–000 (0000)
+
+1.0
+0.9
+0.8
+AM
+0.7
+0.6-
+GGAwithout noise
+GGA,noise=0.05o
+0.5
+GGA.noise = O.1 o
+Gradient.noise=O.lo
+0.4
+101
+102
+BlocksizeSynchrotron intensity with noise = o.1 
+1.0
+0.9
+0.8 -
+AM
+0.7
+0.6
+no noise
+0.5 -
+noisewithoutsmoothing
+noisewithsmoothing
+0.4 
+101
+102
+Blocksize8
+Ho & Lazarian
+is anisotropic. Therefore, the magnetic field direction could be ob-
+tained from the elongated direction of elliptical shape structure after
+the observational map processed by the CFA analysis (Esquivel &
+Lazarian 2005). The elongation depends on the relative importance
+of the three basic MHD modes in turbulence (Kandel et al. 2017). It
+was applied to both observation and simulation data in Yuen et. al
+(2019). However, the study showed that the tracing power of the CFA
+is weaker and the technique is less stable than the gradient technique.
+In this section, we explore the behavior of CFA with the gradient
+amplitude maps.
+A detailed study was conducted to compare the performance be-
+tween gradient and other magnetic field tracing method, including
+CFA (Yuen et. al 2019). One of the issues of CFA showed from Yuen
+et. al (2019) is that the performance of CFA is not stable for the ve-
+locity centroid map. The anisotropy is changed when one selects a
+different block size (For example, figure 15 in Yuen et. al (2019)).
+This change of anisotropy could change 90 degrees by switching the
+block size while the mean field’s direction stays the same throughout
+the region. We repeated this study and extended it to the comparison
+between observable map and gradient amplitude processed map.
+Figure 6 shows how the shape of anisotropy of both maps is
+changed when one selects a different size of a averaging block.
+For sub-Alfvenic simulations like H3S, the mean magnetic field
+strength and direction remain the same throughout the region. For
+CFA, showed from the top side of the figure, we get the same conclu-
+sion as in Yuen et. al (2019). While switching to the small size block
+region, the resolution problem can not only distort the shape of the
+anisotropies in different scales but also destroy the prominent ellipti-
+cal shape. The shape of the elliptical structure is being destroyed for
+the block size is smaller than 120. Also, the direction of anisotropy
+changes when the block size changes.
+However, the situation improves dramatically with the application
+of the GA technique. For the procedure of processing GA-CFA, it is
+same as the computation of the CFA from Yuen et. al (2019) but
+switching the input map to the gradient amplitude map. The bottom
+side of the figure shows the elliptical shape of CFA can be recovered
+after the GA technique. Nonetheless, the anisotropy stays towards
+horizontal direction throughout different block size. From the figure,
+We noticed that there are differences between anisotropy direction
+and magnetic field in block size of 302 but the anisotropy aligns with
+the magnetic field once increase the block size to 602. On the other
+hand, one should mention that the size of the elliptical structure
+is smaller and more elongated compared to the normal CFA. The
+ellipse’s shape exists on a small scale, about 20 to 60 pixels for GA-
+CFA, while it is about 40 to 60 pixels for the CFA. This is due to the
+map process after the gradient amplitude, the morphology of the map
+becomes more filamentary. The size of the filamentary structure is
+more prominent on a small scale in the CFA analysis. So, to improve
+the tracing power of GA-CFA, we have to measure the direction of
+anisotropy on a smaller scale.
+As the performance of CFA improved after combining with the
+GA technique, we then test the improvement of the new GA-CFA
+technique compared to gradient and GGA. We repeat the test showed
+in Figure 1 and extend it to both GGA and GA-CFA technique.
+Inspired by the result from figure 4 and figure 6, we observed a block
+size of 722 would be a common "sweet spot" for both technique
+between the resolution required and the alignment improvement. We
+then pick the sub-block size of 722 for the comparison. The algorithm
+of determining the anisotropy direction of the CFA technique is the
+same as mentioned in the Yuen et. al (2019). For direct comparison
+with Yuen et. al (2019), we also adopt the same pixel distance of
+10 pixels from the center of the elliptical structure for anisotropy
+contour detection.
+Figure 7 shows the results. One can see a significant advan-
+tage of GGA compared to the other two in the figure in terms of
+the AM. For the performance of GGA in both
+synthetic obser-
+vation maps, the AM decreases according to the Alfvenic Mach
+number. The performance drop is mild for GGA for the amount of
+Δ𝐴𝑀 = 𝐴𝑀𝑀𝐴=0.13 − 𝐴𝑀𝑀𝐴=1.17 ∼ 0.1 when 𝑀𝐴 change from
+sub-Alfvenic to super-Alfvenic. The performance of the GA-CFA
+line between the gradient and GGA but closer to GGA in most of
+the cases but with a small effect of fluctuations. Compared to the
+gradient, GA-CFA has a noticeable better performance, which AM
+improves by about 0.1 for most cases. This shows the performance
+of CFA can be improved by unitizing the Gradient amplitude tech-
+nique. The synergy of the gradients and the GA-CFA approach will
+be explored elsewhere.
+7 DISCUSSIONS
+7.1 Connection to earlier gradient studies
+The gradient research opens a new avenue of studying magnetic
+fields and turbulence properties and it is based on of the modern un-
+derstanding of MHD turbulence. Starting from the velocity centroids
+gradient in González-Casanova & Lazarian (2017), studies employed
+later the gradient to different observable maps, such as synchrotron
+intensity/polarization (Lazarian et al. 2017), channel maps (Lazarian
+& Yuen 2018a). This enabled to trace the magnetic field in different
+media from the molecular cloud on the scale of 0.1 pc to the galaxy
+clusters in the scale of 10kpc (see Hu et al. (2020, 2021)). The appli-
+cability of gradient techniques covers two different hydrodynamics
+regimes to both sub-sonic to supersonic regimes. Meanwhile, the
+relationship between gradient and fundamental properties(such as
+𝑀𝑆, 𝑀𝐴, and MHD modes) of MHD turbulence is being discovered.
+The gradient behavior could change 90 degrees in the particular re-
+gion, for instance, shock or fast mode dominated region. In those
+regions, the direction of rotated gradient vectors would change from
+parallel to perpendicular to the magnetic field, which identifies as an
+orthogonal gradient region. Those orthogonal gradients could lower
+the tracing performance of gradient techniques.
+This paper revisits the performance of gradient techniques in the
+sub-sonic turbulent environment. We study the change of perfor-
+mance of different gradient techniques in sub-sonic medium regard-
+ing Alfvenic number systematically. In addition, we extend the study
+of gradient amplitude. This new technique showed a good perfor-
+mance in removing the distortion in the VGT-produced magnetic
+field maps arising from the effects of the intermittent regions dom-
+inated by fast MHD mode. This GGA technique was demonstrated
+to be capable of removing distortions caused by the fast mode. We
+noticed that GGA amplifies the small scale fluctuation that aligned
+with magnetic field, which suppresses the dominance of fast mode.
+However, its performance is influenced by the noise and the relia-
+bility of GGA could drop rapidly in the presence of the noise. We
+showed in section 5 that the GGA performance in the presence of
+noise could be improved by employing suitable Gaussian filtering.
+This enables the new technique to be applied to realistic observation
+data.
+Furthermore, we explore a new way of combing the gradient am-
+plitude maps with the CFA technique in section 6. The classical CFA
+technique has its limitations while applied to the small block size
+region. This results in the requirement of block size > 1002 and
+MNRAS 000, 000–000 (0000)
+
+9
+[t]
+Figure 6. Variation of correlation function anisotropy shapes with respect to block size. The subplot located at the right panel is the magnified view of the centre
+part of the plot. The blue arrow at the upper left plot shows the direction of the plane of sky magnetic field.
+Top panel: CFA
+Bottom panel: GA-CFA
+[t]
+Figure 7. Left panel: Result of Synchrotron intensity map betwen the gradient, GGA and GA-CFA. Right panel: Result of Centroid map betwen the gradient,
+GGA and GA-CFA. X-axis: Alfvenic Mach Number 𝑀𝐴, y-axis: AM.
+MNRAS 000, 000–000 (0000)
+
+Block size = 30
+Block size = 60
+Block size = 120
+Block size = 480
+BposSynchrotron Intensity
+Centroid
+1.0
+0.8
+0.6 
+AM
+0.4 
+0.2 -
+Gradient
+GA-CFA
+GGA
+0.0 
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+MA
+MA10
+Ho & Lazarian
+limits the abilities of the CFA in tracing magnetic field. The new
+combined GA-CFA technique minimizes the block size e.g. to 302
+without decreasing the performance of the technique. This extends
+the applicability of the CFA technique and makes it competitive to
+the gradient technique.
+7.2 Intensity structure and velocity caustics in Channel Map
+The theory of describing the fluctuations of intensity within spectro-
+scopic data that arise from turbulence was formulated in (Lazarian
+& Pogosyan 2000). There the concept of velocity caustics has been
+proposed to describe the effect of turbulent velocities eddies made to
+the channel map. This provided the basis for the technique of tracing
+magnetic fields using velocity channel gradients.
+However, the density also affect fluctuations in channel map fluc-
+tuations. Several HI studies have been discussed on the influence of
+thermal broadening of warm phase made to channel map and the
+importance of cold phase media. The applicability of Lazarian &
+Pogosyan (2000) to galactic HI was questioned in (Clark et al. 2019).
+A rebuttal to these arguments was given by Yuen et. al (2019) and
+the applicability of the LP00-based approach was demonstrated in
+Yuen et al. (2021) where the Velocity Decomposition Algorithm was
+introduced to deal with density fluctuations in subsonic flow. The
+later observational study by Yuen et al. (2022) reported the velocity
+caustics could be fully restored after applying the algorithm.
+Our study in section 4.3 showed provides another argument in
+favor of the applicability of the LP00 theory to multi-phase media.
+We showed that if the phases of the media move together in the
+galactic disk, they can be viewed as a unified turbulent system, and
+our result from figure 3 suggests that most of the information of
+velocity anisotropy can be preserved without the VDA.
+8 SUMMARY
+This paper extends our studies the Gradient Technique (GT) in the
+sub-sonic environment. Our main results are:
+1. The alignment between gradient and POS magnetic field is
+better in the subsonic regimes compared to the supersonic one.
+2. In the multi-phase media, the morphology of filamentary struc-
+ture in the channel map and the statistical anisotropy of thin channel
+intensity fluctuations is preserved in the presence of thermal broad-
+ening if the phases are moving together.
+3. We extended the study of GGA introduced in the Ho & Lazar-
+ian (2021). We examined the applicability of GGA in the synthetic
+observation map with noise added. The performance of GGA is sen-
+sitive to noise, but the employment of the Gaussian kernel alleviates
+the noise effect.
+4. We demonstrated that the gradient amplitude maps can be suc-
+cessfully combined with Correlation Function Analysis (CFA). In
+this case the anisotropy can is prominent in small block of the order
+of 302. This makes the new technique competitive with the gradient
+technique.
+MNRAS 000, 000–000 (0000)
+
+11
+ACKNOWLEDGEMENTS
+We acknowledge Ka Ho Yuen and Yue Hu for the fruit-
+ful
+discussions.
+We
+acknowledge
+the
+support
+the
+NASA
+ATP
+AAH7546
+and
+NASA
+TCAN
+144AAG1967
+grants.
+SOFTWARE
+Julia-v1.2.0/Julia-v1.8.2, Jupyter/miniconda3, LazTech-VGT (Yuen
+& Lazarian 2017a) : https://github.com/kyuen2/LazTech-VGT
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
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+Springer Nature 2021 LATEX template
+Resilient Model Predictive Control of
+Distributed Systems Under Attack Using
+Local Attack Identification
+Sarah Braun1*, Sebastian Albrecht1 and Sergio Lucia2
+1*Siemens AG, Otto-Hahn-Ring 6, 81739 M¨unchen, Germany.
+2TU Dortmund University, August-Schmidt-Straße, 44227
+Dortmund, State.
+*Corresponding author(s). E-mail(s): [email protected];
+Contributing authors: [email protected];
+[email protected];
+Abstract
+With the growing share of renewable energy sources, the uncertainty
+in power supply is increasing. In addition to the inherent fluctuations
+in the renewables, this is due to the threat of deliberate malicious
+attacks, which may become more prevalent with a growing number
+of distributed generation units. Also in other safety-critical technology
+sectors, control systems are becoming more and more decentralized,
+causing the targets for attackers and thus the risk of attacks to
+increase. It is thus essential that distributed controllers are robust
+toward these uncertainties and able to react quickly to disturbances
+of any kind. To this end, we present novel methods for model-based
+identification of attacks and combine them with distributed model pre-
+dictive control to obtain a resilient framework for adaptively robust
+control. The methodology is specially designed for distributed setups
+with limited local information due to privacy and security reasons. To
+demonstrate the efficiency of the method, we introduce a mathematical
+model for physically coupled microgrids under the uncertain influence
+of renewable generation and adversarial attacks, and perform numeri-
+cal experiments, applying the proposed method for microgrid control.
+Keywords: Attack Identification, Robust Nonlinear Control, Distributed
+Model Predictive Control, Microgrids Under Attack
+1
+arXiv:2301.05547v1  [cs.SY]  13 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+1 Introduction
+Due to the energy transition, power generation is facing a technological
+change toward increasingly distributed generation, primarily from renewable
+energy sources. Also in other technology areas such as industrial production
+or the transport sector, advancing automation and digitization are creating
+an increasing need for distributed control methods that can be applied to
+safety-critical systems in real time. When designing such methods, it is impor-
+tant to take into account that distributed systems with many components can
+increase flexibility, but at the same time provide many targets for malicious
+attacks. Therefore, distributed control methods should be designed robustly
+and securely, and complemented with appropriate tools to increase the sys-
+tem’s resilience to any type of disruption, which is particularly challenging in
+the event of unpredictable, adversarial attacks.
+Model predictive control (MPC) is one of the most popular control methods
+for dynamic systems in various fields of application as it applies to multivari-
+able systems and allows to include constraints and cost functions in a natural
+way. Based on updated measurements, it repeatedly computes optimal inputs
+to the system at each sampling time. Distributed MPC (DMPC) methods, see
+[1] for an overview and [2] for security-related DMPC, are designed for large
+systems of coupled subsystems and locally apply MPC in each subsystem. In
+contrast to fully decentralized approaches where the neighbors’ dynamic evo-
+lution is unknown to every subsystem, DMPC schemes involve some exchange
+of information among neighbors. In [3], e.g., subsystems provide each other
+with corridors in which future values of their coupling variables lie. Given
+such information about the uncertainty range, robust MPC can be applied
+to explicitly take uncertain influences into account when computing optimal
+inputs. Robust MPC schemes typically build upon tube-based ideas as in [4] or
+multi-stage approaches [5]. It has been demonstrated in several works [6, 7, 8]
+that robust (D)MPC cannot only be applied for robustness against uncertain
+parameters or neighboring couplings, but also against adversarial attacks.
+While robust MPC can reduce the impact of disruptions if the uncertainty
+ranges are known, appropriate security measures for unknown attacks require
+that their presence and points of attack are recognized in the first place. In
+this context, Pasqualetti et al. [9] introduce attack detection and identification
+(ADI) as the tasks of revealing the presence of an attack and localizing all
+attacked system components. For both linear and nonlinear dynamics, there
+are many methods to detect and identify attacks or, closely related, unin-
+tentional system faults. For a broad overview of physics- and control-based
+approaches we refer to the survey in [10]. Some works like [9, 11, 12] design
+unknown-input observers and employ one observer per attack scenario for iden-
+tification, resulting in a combinatorial complexity. Moreover, works on fault
+identification [11] often assume that all possible faults are known, which is an
+invalid assumption for adversarial attacks. In distributed ADI, each subsys-
+tem employs its own estimator to detect and identify local perturbations, be
+it based on observer systems as in [11, 12, 13] or sparse optimization problems
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+3
+as in [14]. To represent the influence of other subsystems, the local problems
+typically involve measurements of the neighboring couplings transmitted by
+the neighbors [11] or approximated by adaptive local estimators [13].
+In recent years, several approaches that intertwine the handling of attacks
+with (robust) DMPC have been published. In [6], e.g., a DMPC-based strategy
+is presented by which systems reach resilient consensus even if some agents are
+malicious and transmit disturbed state values to their neighbors. An attack
+identification method using Bayesian inference is introduced in [15] and com-
+bined with DMPC to solve robust chance-constrained problems. The approach
+involves testing a series of hypotheses about the attack set and requires full enu-
+meration of all possible attack scenarios. To avoid the resulting combinatorial
+complexity, we combined a DMPC scheme from [3] with our optimization-
+based global ADI method from [16] and proposed an adaptively robust DMPC
+method in [17] for targeted robust control against previously identified attack.
+The contribution of this work, which is an extension of [18], consists in two
+novel approaches for distributed attack identification, a DMPC scheme embed-
+ding these ADI methods for adaptively robust control, and a numerical case
+study to illustrate the proposed resilient control framework using an example
+of interconnected microgrids under attack. The new methods for model-based
+distributed ADI are derived in Section 3 (significantly more detailed compared
+to [18] and including one completely new method). They involve a targeted
+exchange of information between neighbors and solve sparse optimization prob-
+lems to locally identify an attack. The identified insights are used by the DMPC
+framework for adaptively robust control presented in Section 4 (considerably
+exceeding the summarized version in [18]) to initiate suitable preparatory
+measures against previously identified attacks. Unlike the related technique
+introduced in [17], it involves one of the new distributed ADI techniques pre-
+sented in this paper. Finally, we introduce here a more detailed numerical case
+study (in comparison to [18]) with a nonlinear dynamic model for tertiary
+control of interconnected microgrids under attack in Section 5 and perform
+numerical experiments with several attack scenarios in Section 6, illustrating
+the great potential of our resilient control framework for attacked microgrids
+with uncertain renewable generation.
+2 Problem Formulation
+We consider nonlinear dynamic systems with states x ∈ X ⊆ Rnx, inputs
+u ∈ U ⊆ Rnu, outputs y ∈ Y ⊆ Rny, and uncertain parameters w ∈ W ⊆ Rnw
+that behave according to discrete-time dynamics of the form
+xk+1 = f
+�
+xk, uk + ak, wk�
+,
+yk+1 = c
+�
+xk+1�
+,
+(1)
+with nonlinear functions f : X×Rnu ×W → X and c : X → Y that are assumed
+to be sufficiently smooth. The system is exposed to the threat of potential
+
+Springer Nature 2021 LATEX template
+4
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+attacks, which are modeled by attack inputs a ∈ A(u) ⊆ Rnu unknown to the
+controller. We consider arbitrary attack vectors a and make no assumptions
+about the set A(u) of possible attacks. While the attack model is additive
+in the input, an attack a affects the states and outputs of the system in a
+nonlinear, nonadditive way.
+The system is partitioned into a set D of subsystems I with local states
+xI ∈ XI ⊆ RnxI , local control inputs uI ∈ UI ⊆ RnuI , local attack inputs
+aI ∈ AI(u) ⊆ RnaI , local outputs yI ∈ YI ⊆ RnyI , and uncertain parameters
+wI ∈ WI ⊆ RnwI . A distributed version of the dynamic system in (1) with
+local dynamic functions fI and local output functions cI is formulated as
+xk+1
+I
+= fI
+�
+xk
+I, uk
+I + ak
+I, �zk
+NI, wk
+I
+�
+,
+zk+1
+I
+= hI
+�
+xk+1
+I
+�
+,
+yk+1
+I
+= cI
+�
+xk+1
+I
+�
+,
+(2)
+where the physical interconnection of subsystems is modeled through coupling
+variables zI ∈ ZI ⊆ RnzI that are related to the local states xI through local
+coupling functions hI : XI → ZI. Since the dynamic evolution of the neigh-
+boring coupling variables zNI(t) during some time interval t ∈ [tk, tk+1] is not
+determined by subsystem I, distributed models typically approximate zNI(t)
+using some information provided by the neighbors. Here, we apply a parame-
+terization scheme proposed in [19] and represent zI(t) on [tk, tk+1] as the linear
+combination
+zI(t) =
+�n
+�
+j=1
+zk,j
+I βk
+j (t)
+of �n basis functions βk
+1, . . . , βk
+�n
+:
+[tk, tk+1)
+→
+R. The coupling coeffi-
+cients zk,j
+I
+are exchanged among neighbors and �zk
+I denotes the coefficient
+matrix �zk
+I := (zk,1
+I
+, . . . , zk,�n
+I
+) ∈ �ZI ⊆ RnzI ×�n. For a simplified notation, we
+introduce the chained local coupling function ζI := hI◦fI and the chained local
+output function ηI := cI ◦ fI. Similarly, the dense output coupling function
+�ζI : XI × Rnu × �ZNI × WI → �ZI maps to the space �ZI of coupling coefficients.
+Based on the local coupling functions ζI, so-called nominal coupling
+values ¯zk
+I can be determined for the undisturbed case of no attack:
+¯zk+1
+I
+:= ζI
+�
+xk
+I, uk
+I,�¯z
+k
+NI, 0
+�
+.
+(3)
+This nominal value is attained if no local attack is applied to the system, i.e.,
+ak
+I = 0, no model uncertainty is present, i.e., wk
+I = 0, and all neighboring
+subsystems also behave according to their nominal values, i.e., �zk
+NI = �¯z
+k
+NI. For
+all methods presented in this paper we assume:
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+5
+Assumption 1 At each time k, each subsystems I ∈ D transmits the predicted
+nominal values �¯zk
+I , . . . ,�¯zk+Np−1
+I
+of its coupling coefficients with prediction horizon
+Np ∈ N to its neighbors.
+Given this exchange of information among neighbors, the above definition
+in (3) allows for a distributed calculation of the nominal values in a receding
+horizon fashion, where the local values computed and transmitted by subsys-
+tem I at time k are used by its neighbors to update their predictions one time
+step later. The definition further requires suitable initial values �¯z
+0
+I to be avail-
+able. For simplicity, we assume the system to be in a steady state x0 at time 0
+and take ¯z0,j
+I
+= hI(x0
+I) for all j ∈ {1, . . . , �n}.
+Finally, each subsystem is subject to a set of local constraints
+gI
+�
+xk
+I, uk
+I + ak
+I, �zk
+NI, wk
+I
+�
+≤ 0
+(4)
+for some nonlinear function gI : XI × RnuI × �ZNI × WI → RngI that must be
+satisfied at all times.
+3 Distributed Attack Identification Based on
+Sparse Optimization
+The goal of this section is to propose a distributed ADI method that, in con-
+trast to global methods, does not involve a central authority which has access
+to a global model of the system. Instead, we formulate a bank of local problems
+that allow each subsystem to identify a suspicion a∗
+I about a potential local
+attack aI based on locally available model knowledge and, possibly, interaction
+with its neighboring subsystems. In contrast to the centralized ADI method
+we presented in [16], no local model knowledge is published globally.
+Before that, we briefly recall the distributed method for the detection of
+attacks that has already been presented in [16]. It is based on each subsystem I
+monitoring the deviations ∆zk+1
+I
+:= zk+1
+I
+− ¯zk+1
+I
+in its local coupling variables
+from the respective nominal values ¯zk+1
+I
+. As the nominal values ¯zk+1
+I
+defined
+in (3) are attained in the undisturbed case, a deviation from them indicates
+a disturbance at time k. Using a detection threshold τD ∈ R>0, the method
+detects an attack if ∥∆zk+1
+I
+∥∞ > τD for any I, i.e., if a distinct deviation is
+observed in any subsystem. To ensure that only significant attacks are revealed
+rather than small model inaccuracies or measurement noise, one can assume
+a probability distribution of the uncertainty and define τD accordingly as in,
+e.g., [11]. Even if subsystem I detects an attack by observing a clear deviation
+∥∆zk+1
+I
+∥∞ > τD, it does not necessarily have to be caused by an attack ak
+I ̸= 0
+in I, but can just as well be caused by neighboring subsystems deviating from
+their nominal couplings �¯z
+k
+NI. Identifying the root of the disturbance and thus
+locating the attack is the task of attack identification.
+
+Springer Nature 2021 LATEX template
+6
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+I
+L
+K
+Local ADI
+Local ADI
+Local ADI
+involving
+problem (5)
+�¯z
+k
+I, ∆�zk
+I
+�¯z
+k
+K, ∆�zk
+K
+�¯z
+k
+K, ∆�zk
+K
+�¯z
+k
+L, ∆�zk
+L
+Fig. 1: If neighboring subsystems in a distributed system exchange suitable
+information about their local coupling variables, each subsystem can employ
+a local ADI method to identify suspicions about unknown local attack inputs.
+In this paper, also the identification of attacks is addressed in a distributed
+manner. Depending on the amount and type of information that neighbors
+are willing to share, we derive two different versions of local identification
+problems. Clearly, the more specific the transmitted information describes
+the neighbors’ behavior, the more precisely a local attack or even an attack
+on neighboring subsystems can be identified. Therefore, the design of a local
+identification problem needs to suitably balance the required amount of infor-
+mation and the significance of the obtained suspicions. For the first local
+identification problem that we establish, we propose that in addition to the
+exchange of nominal values �¯z
+k
+I according to Assumption 1, also the deviations
+∆�zk
+I in the coupling coefficients are repeatedly transmitted to neighboring sub-
+systems. This exchange is performed at each step k when an attack is detected
+and is illustrated in Figure 1. Assuming that each subsystem can locally mea-
+sure the impact onto its output variables yk+1
+I
+∈ YI ⊆ RnyI , we formulate a
+local attack identification problem to identify local attacks ak
+I as
+min
+aI
+∥aI∥1
+s.t.
+���yk+1
+I
+− ηI
+�
+xk
+I, uk
+I + aI,�¯z
+k
+NI + ∆�zk
+NI, 0
+����
+2 ≤ εI.
+(5)
+A solution of problem (5), which has already been proposed in [18], identifies
+a local suspicion a∗
+I for some subsystem I, which is ℓ1-norm sparsest among all
+possible attack vectors in RnuI that explain the observed output yk+1
+I
+accord-
+ing to the local model with output function ηI up to a predefined tolerance
+εI ∈ R≥0, neglecting possible parametric uncertainties wk
+I . While the opti-
+mization variable aI ∈ RnuI represents the unknown attack to be identified,
+the local state xk
+I, input uk
+I, and output yk+1
+I
+are measured or known from
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+7
+local control computations, and the values �¯z
+k
+NI and ∆�zk
+NI, and thus the actual
+neighboring coupling values �zk
+NI = �¯z
+k
+NI + ∆�zk
+NI, are transmitted by neighbors.
+Computing a sparse suspicion to identify the attack is common in related work
+on attack identification, e.g., [9, 14] and is justified by the observation that
+attackers typically have limited resources and are thus confined to impairing
+only few control components. Some approaches formulate related optimization
+problems using an ℓ0-“norm” cost term ∥aI∥0 to count the number of attacked
+inputs, but solving them requires solution methods from mixed integer pro-
+gramming and is NP-hard [9]. To reduce the computational complexity and
+to obtain a numerically more tractable problem, the ℓ0-“norm” is typically
+relaxed by the ℓ1-norm, see also [16, 20].
+If the neighboring subsystems in NI agree to provide I with even more
+information, subsystem I can apply another version of local identification
+problem, which allows to draw not only conclusions about a potential local
+attack ak
+I, but even about attack inputs ak
+NI in the neighborhood of I. Since
+distributed methods are often applied when sensitive local information must
+not be made publicly available, we assume that neighbors still seek to keep
+their analytical model knowledge private and are only willing to reveal suitable
+numerical derivative information evaluated at the current iterate. We pursued
+a similar approach for the centralized ADI method presented in [16], involv-
+ing the exchange of locally computed sensitivity matrices. To motivate which
+kind of sensitivity information about the dynamic behavior of its neighbors
+subsystem I requires, we approximate the neighboring influence onto the local
+output yI by a first-order Taylor expansion of ηI(xk
+I, uk
+I + ak
+I, �zk
+NI, 0) in the
+�zNI-argument around the nominal value �¯z
+k
+NI. To this end, we define a local
+sensitivity function Sz
+INI : RnuI → RnyI ×nzNI , which maps each given attack
+input aI ∈ RnuI to the Jacobian
+Sz
+INI (aI) := ∂ηI
+∂�zNI
+�
+xk
+I, uk
+I + aI,�¯z
+k
+NI, 0
+�
+,
+that expresses the first-order dependence of the local output function ηI on
+the neighboring coupling variables �zNI. It can be evaluated locally by I and
+allows to approximate the local output variables yk+1
+I
+according to Taylor’s
+theorem, e.g., [21, §7] as
+yk+1
+I
+= ηI
+�
+xk
+I, uk
+I + ak
+I,�¯z
+k
+NI, 0
+�
++ Sz
+INI
+�
+ak
+I
+�
+∆�zk
+NI + Rlin
+I
++ Rw
+I .
+(6)
+Here, the remainder term of the Taylor expansion is denoted by Rlin
+I
+and can be
+estimated similar to the upper bound proven in [16]. The term Rw
+I represents
+a model error which occurs as all uncertain parameters wk
+I are considered zero
+in (6) and due to the fact that the distributed model in (2) only approximates
+the global dynamics in (1).
+
+Springer Nature 2021 LATEX template
+8
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+At this point, the additional sensitivity information provided by the neigh-
+bors NI of I comes into play. Denoting the coupling coefficients of the
+neighbors’ neighbors by �zNNI , we introduce two types of sensitivity matrices as
+�Sa
+NI := ∂�ζNI
+∂aNI
+�
+xk
+NI, uk
+NI,�¯z
+k
+NNI , 0
+�
+and
+�Sz
+NI := ∂�ζNI
+∂�zNNI
+�
+xk
+NI, uk
+NI,�¯z
+k
+NNI , 0
+�
+.
+The function �ζNI denotes the dense coupling function of all neighbors in NI,
+which maps to the space �ZNI of coupling coefficients �zNI and is obtained
+by combining the local dense coupling functions �ζL for all L ∈ NI. Hence,
+the sensitivity matrices �Sa
+NI and �Sz
+NI represent first-order approximations of
+how disturbances in uNI and �zNNI affect the coupling coefficients �zNI. If the
+neighbors in NI provide subsystems I with this information, the deviation
+∆�zk
+NI of neighboring couplings �zk
+NI from their transmitted nominal values �¯z
+k
+NI
+can be expressed as
+∆�zk
+NI = �Sa
+NIak
+NI + �Sz
+NI∆�zk
+NNI + Rlin
+NI + Rw
+NI.
+(7)
+The model error Rw
+NI is caused by the uncertain influence of the parameters
+wk
+NI and the linearization error Rlin
+NI denotes the Taylor remainder term when
+expanding the neighbors’ coupling function �ζNI around �¯z
+k
+NNI . The represen-
+tation in (7) gives subsystem I more detailed insights into why its neighbors’
+coupling values �zk
+NI differ from the nominal values �¯z
+k
+NI. More precisely, it
+allows subsystem I to distinguish whether the deviation is caused by an attack
+ak
+NI that the neighbors are exposed to or whether they pass on the disturbing
+effect of any of their neighbors. In order to figure out which source of distur-
+bance applies, subsystem I solves the following local identification problem
+with optimization variables aI, aNI, and ∆�zNNI :
+min
+aI,aNI ,∆�zNNI
+∥aI∥1 + ∥aNI∥1 +
+���∆�zNNI
+���
+1
+s.t.
+���yk+1
+I
+− ηI
+�
+xk
+I, uk
+I + aI,�¯z
+k
+NI, 0
+�
++ Sz
+INI(aI)
+�
+�Sa
+NIaNI + �Sz
+NI∆�zNNI
+� ���
+2 ≤ εI.
+(8)
+An optimal solution (a∗
+I, a∗
+NI, ∆�z∗
+NNI ) of problem (8) is sparsest with respect
+to the ℓ1-norm among all feasible points satisfying the constraints, which are
+obtained by combining (6) and (7) and neglecting all error terms. Similar
+to problem (5), the constraints are relaxed by some tolerance εI ∈ R≥0 to
+account for model inaccuracies. Besides the local quantities uk
+I, yk+1
+I
+, and
+xk
+I, which are known, measured, or estimated by the local control scheme,
+problem (8) also involves the nominal coefficients �¯z
+k
+NI, which are assumed
+to be exchanged among neighboring subsystems according to Assumption 1.
+Instead of the coupling deviations ∆�zk
+NI, the exchange of which is illustrated
+in Figure 1 and taken for granted by the first local identification problem
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+9
+Algorithm 1 Distributed Attack Detection and Identification Based on Sparse
+Optimization
+Input: local dynamic model for each subsystem I ∈ D as in (2),
+version ∈ {1, 2}
+1: detected = false, a∗
+I = 0 for all I
+▷ initialization
+2: for I ∈ D do
+▷ distributed attack detection
+3:
+measure zI, determine ∆zI
+4:
+if ∥∆zI∥∞ > τD then
+5:
+detected = true
+6:
+break
+7:
+end if
+8: end for
+9: if detected then
+▷ distributed attack identification
+10:
+for I ∈ D do
+11:
+if version == 1 then
+12:
+obtain coupling deviation ∆�zNI from neighbors
+13:
+solve local identification problem (5) to obtain a∗
+I
+14:
+else
+15:
+obtain sensitivity information �Sa
+NI, �Sz
+NI from neighbors
+16:
+solve local identification problem (8) to obtain a∗
+I
+17:
+end if
+18:
+end for
+19: end if
+20: return detected, a∗
+I for all I
+(5), the new distributed ADI approach requires all neighbors to provide the
+sensitivity matrices �Sa
+NI and �Sz
+NI. The third sensitivity matrix Sz
+INI (aI) that
+is contained in the constraints of problem (8), in contrast, is computed locally
+by subsystem I in dependence on the optimization variable aI.
+Now that two different formulations of local identification problems have
+been presented, we briefly explain how a complete distributed ADI method is
+obtained from the local optimizations problem (5) or (8), respectively, summa-
+rized as Algorithm 1. The distributed detection scheme is based on monitoring
+the coupling variables and raises an alarm if an abnormal deviation ∆zI > τD
+is observed in any subsystem I. Then, the identification procedure is initiated
+and neighboring subsystems exchange the necessary information to set up the
+identification problem (5) or (8), depending on which version is applied, and
+compute a solution to obtain a suspicion a∗
+I of the local attack. If problem (8)
+is considered, the solution also suggests suspicions a∗
+NI and ∆�z∗
+NNI about the
+disturbing activities in the neighborhood.
+Since the problem formulations in (5) and (8) show some similarities to the
+global identification problem of our publication [16], some of the theoretical
+considerations in [16] can be adopted with only minor changes. E.g., an upper
+bound on the remainder term of the Taylor expansion can be obtained for the
+
+Springer Nature 2021 LATEX template
+10
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+linearization error Rlin
+I
+in (6), when adapting the reasoning of [16] to the fact
+that here the expansion is only applied in the �zNI-argument but not the input.
+The major difference between the identification problems for global versus
+distributed ADI is, however, that the constraints in problem (5) and (8) are
+nonlinear, whereas a linear problem is considered in [16]. As a consequence, the
+theoretical results from [20] on relaxing the ℓ0-“norm” cost term in compressed
+sensing problems by the ℓ1-norm are not applicable here since Candes and Tao
+restrict their considerations to linear constraints. In fact, there is a body of
+research on nonlinear compressed sensing, e.g., [22, 23], the results of which
+can be useful to prove rigorous guarantees for the distributed ADI method
+presented in this section. However, a precise elaboration of such proofs is out
+of scope for this paper and a promising direction for future work.
+4 Resilient Distributed MPC
+While methods for attack identification are a very powerful tool to localize
+a priori unknown attacks and thus improve the resilience of control systems
+under malicious disturbances, they cannot prevent future attacks or reduce
+their impact. On the other hand, robust control schemes can limit the impact
+of a perturbation by ensuring that no constraints are violated, but require
+information about the value range in which possible disturbances will lie, which
+is typically not available for unknown adversarial attacks. We combine the
+advantages of both approaches by embedding the proposed ADI method into
+a DMPC setup, thus utilizing the identified insights about the attacker toward
+targeted robust DMPC. To this end, we first describe an existing approach for
+robust DMPC in Section 4.1, and enhance it with Algorithm 1 to obtain an
+adaptively robust DMPC scheme in Section 4.2 that computes robust control
+inputs against previously identified attacks in a distributed manner.
+4.1 Contract-Based Robust Distributed MPC
+By robust control, we refer to computing control inputs that ensure all con-
+straints to a system with uncertain influences being met in all possible cases.
+In [5], Lucia et al. introduce a multi-stage scheme for robust nonlinear MPC
+(NMPC), which considers discrete sets of scenarios and represents the possible
+evolution of the system state in a scenario tree like the one shown in Figure 2.
+In a distributed dynamic system, the neighbors’ couplings zNI behave in an
+uncertain way to the eyes of subsystem I, and, therefore, robust MPC can
+also be used to design distributed MPC methods as long as each subsystem
+is provided with information about the range of possible neighboring coupling
+values. In [3], this idea is implemented by Lucia et al. introducing so-called
+contracts ZI, which are corridors containing predicted reachable values of the
+coupling variables zI and are exchanged among neighbors. At time k, the
+reachable state set X l+1,[k]
+I
+of all values that the local state xl+1
+I
+may attain
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+11
+�
+X 1,[0]
+I
+�
+X 2,[0]
+I
+�
+X 3,[0]
+I
+�
+X 4,[0]
+I
+x0
+I
+x1,s7
+I
+x2,s9
+I
+x3,s9
+I
+x4,s9
+I
+x2,s8
+I
+x3,s8
+I
+x4,s8
+I
+x2,s7
+I
+x3,s7
+I
+x4,s7
+I
+x1,s4
+I
+x2,s6
+I
+x3,s6
+I
+x4,s6
+I
+x2,s5
+I
+x3,s5
+I
+x4,s5
+I
+x2,s4
+I
+x3,s4
+I
+x4,s4
+I
+x1,s1
+I
+x2,s3
+I
+x3,s3
+I
+x4,s3
+I
+x2,s2
+I
+x3,s2
+I
+x4,s2
+I
+x2,s1
+I
+x3,s1
+I
+x4,s1
+I
+Fig. 2: A scenario tree as in the multi-stage approach to robust MPC [5], here
+shown for time k = 0 and Np = 4, provides a natural and computationally
+efficient way to approximate the reachable sets X l,[k]
+I
+(indicated in gray) by
+discrete node sets �
+X l,[k]
+I
+(blue) explored by the tree.
+at time l + 1 under all possible uncertainty realizations, is computed as
+X l+1,[k]
+I
+:=
+�
+fI
+�
+xl
+I, ul
+I + al
+I, �zl
+NI, wl
+I
+�
+:
+xl
+I ∈ X l,[k]
+I
+, al
+I ∈ Al,[k−1]
+I
+, �zl
+NI ∈ �
+Zl,[k−1]
+NI
+, wl
+I ∈ Wl,[k−1]
+I
+�
+with X k,[k]
+I
+:= {xk
+I}. From this, the contract Zl,[k]
+I
+for zl
+I at time k is derived as
+Zl,[k]
+I
+:=
+�
+hI
+�
+xl
+I
+�
+: xl
+I ∈ X l,[k]
+I
+�
+.
+Similarly, contracts �
+Zl,[k]
+I
+for the coupling coefficients �zl
+I are obtained using
+the dense coupling function �ζ. These sets are computed locally at time k,
+provided that each subsystem knows attack and parameter uncertainty sets
+Al,[k−1]
+I
+and Wl,[k−1]
+I
+and additionally receives its neighbors’ contracts �
+Zl,[k−1]
+I
+.
+If all these uncertainty sets are discrete or subsystem I chooses finite subsets
+as sample scenarios, it can locally build a scenario tree as in Figure 2. The
+tree contains one node xl,s
+I
+for each time l ∈ {k, . . . , k + Np} with prediction
+horizon Np and each scenario s ∈ Σ[k−1]
+I
+, where Σ[k−1]
+I
+is the finite local index
+set of scenario indices s. The local scenario trees allow to efficiently compute
+finite approximations �
+X l,[k]
+I
+of the reachable sets X l,[k]
+I
+as the set of tree nodes
+xl,s
+I
+that are reached by subsystem I at stage l in any scenario s ∈ Σ[k−1]
+I
+.
+This is indicated by blue shapes in Figure 2 and explained in detail in [8].
+
+Springer Nature 2021 LATEX template
+12
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+Corresponding approximated contracts �
+Zl,[k]
+I
+are obtained as
+�
+Zl,[k]
+I
+:=
+�
+�ζI
+�
+xl,s
+I , ul,s
+I + al,s
+I , �zl,s
+NI, wl,s
+I
+�
+: s ∈ Σ[k−1]
+I
+�
+⊆ �
+Zl,[k]
+I
+and have been proven to work well in practice [8, 17]. Considering every pos-
+sible evolution of the uncertain system for the future time steps k, . . . , k + Np
+according to the finite scenario set Σ[k−1]
+I
+, contract-based DMPC using multi-
+stage NMPC computes robust control inputs uk
+I, . . . , uk+Np−1
+I
+according to the
+following optimal control problem based on the work of Lucia et al. in [3, 5]
+min
+xl,s
+I ,ul,s
+I
+�
+s∈Σ[k−1]
+I
+αs
+I
+k+Np−1
+�
+l=k
+ℓI
+�
+xl,s
+I , ul,s
+I + al,s
+I , �zl,s
+NI, wl,s
+I
+�
+s.t.
+xk,s
+I
+= xk
+I,
+xl+1,s
+I
+= fI
+�
+xl,s
+I , ul,s
+I + al,s
+I , �zl,s
+NI, wl,s
+I
+�
+,
+gI
+�
+xl,s
+I , ul,s
+I + al,s
+I , �zl,s
+NI, wl,s
+I
+�
+≤ 0,
+(9)
+xl+1,s
+I
+∈ XI, ul,s
+I
+∈ UI,
+xl,s
+I
+= xl,s′
+I
+⇒ ul,s
+I
+= ul,s′
+I
+,
+min
+�
+�
+Zl,[k−1]
+I
+�
+≤ �ζI
+�
+xl,s
+I , ul,s
+I + al,s
+I , �zl,s
+NI, wl,s
+I
+�
+≤ max
+�
+�
+Zl,[k−1]
+I
+�
+,
+for all
+s ∈ Σ[k−1]
+I
+, s′ ∈ Σ[k−1]
+I
+, l ∈ {k, . . . , k + Np − 1} .
+An optimal solution of problem (9) provides a set of state trajectories starting
+at xk
+I for all scenarios, behaving according to the local discrete-time dynamics
+as in (2), and taking only feasible states xl+1,s
+I
+∈ XI. The optimal inputs are
+chosen to be feasible, to satisfy the constraints in (4) in all scenarios s ∈ Σ[k−1]
+I
+and at all times l, and to minimize the local costs ℓI weighted over all scenarios
+with weights αs
+I ∈ R≥0. The problem formulation takes into account that
+future control inputs can be adapted when new measurements are available,
+while input values ul,s
+I , ul,s′
+I
+that are applied to the same tree node have to
+coincide because a real-time controller cannot anticipate the future. Finally,
+for consistency, we require each element �zl,s
+I
+of the updated contract �
+Zl,[k]
+I
+to be within the bounds of the previous contract �
+Zl,[k−1]
+I
+. For details on the
+purpose and the theoretical consequences of the last two groups of constraints
+we refer to the original works [3, 5] and our own work [8].
+4.2 Adaptively Robust Distributed MPC
+While we have explained in Section 4.1 how updated contracts �
+Zl,[k]
+I
+are
+calculated at each time k from a solution of problem (9), we have not yet com-
+mented on how to obtain similar scenario sets �
+Al,[k]
+I
+and �
+Wl,[k]
+I
+for unknown
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+13
+attacks al
+I and uncertain parameters wl
+I. For the latter, suitable samples are
+usually provided by forecasts, historical data, or technical properties of the
+system components. For unknown attacks, however, it would be very restric-
+tive to assume that appropriate scenario sets �
+Al,[k]
+I
+are provided. Choosing
+few random attacks as samples as in [8] cannot be expected to achieve sat-
+isfied constraints in all cases, while choosing a very large number of samples
+may cover the set AI of possible attacks sufficiently well, but leads to com-
+putationally intractable problems since the size of the scenario tree grows
+exponentially in the number of scenarios. To address this issue, we proposed a
+more general, adaptively robust MPC approach in [17] that utilizes available
+knowledge about the attackers gained from attack identification to design the
+sets �
+Al,[k]
+I
+and is repeated in this section. Unlike in [17], here the distributed
+ADI approaches from Section 3 are embedded in a DMPC setup, resulting in a
+fully distributed control framework that does not require any central instance.
+The approach has already been described in [18] and is presented here in
+further depth.
+The method is designed for local attacks aI that follow a probability distri-
+bution with unknown, time-invariant expected value µI ∈ RnuI and standard
+deviation σI ∈ R
+nuI
+≥0 . The basic idea is to repeatedly estimate these parame-
+ters at each time k based on the solutions a∗,l
+I
+of the local attack identification
+problem at previous times l ≤ k, and to adapt the uncertainty sets �
+Al,[k] for
+possible attacks al accordingly. More precisely, at time k the mean µ[k]
+I
+and
+sample standard deviation σ[k]
+I
+of all previously identified values a∗,l
+I
+given as
+µ[k]
+I
+:=
+1
+k + 1
+k
+�
+l=0
+a∗,l
+I
+and
+σ[k]
+I
+:=
+�
+1
+k
+k
+�
+l=0
+�
+a∗,l
+I
+− µ[k]
+I
+�2
+� 1
+2
+(10)
+serve as estimates for µI and σI. According to the local identification results
+until time k, the uncertainty of possible attacks al
+I for future time steps l is
+represented by three scenarios for each component (ak
+I)i for i ∈ {1, . . . , nuI}
+�
+Al,[k]
+I
+=
+�
+i∈I
+�
+µ[k]
+i , µ[k]
+i
++ σ[k]
+i , µ[k]
+i
+− σ[k]
+i
+�
+.
+(11)
+The combination of contract-based robust DMPC from Section 4.1 and the dis-
+tributed ADI method from Section 3 results in an adaptively robust distributed
+MPC method that is summarized in Algorithm 2.
+We formulate Algorithm 2 involving the local identification problem (5)
+and thus the first version of Algorithm 1 since this is what we apply in the
+numerical experiments presented in Section 6. Clearly, Algorithm 2 can also
+be defined based on the second version of Algorithm 1 solving problem (8).
+In this case, subsystem I can additionally modify the transmitted contracts
+�
+ZNI in such a way that the locally identified suspicions a∗
+NI, ∆�z∗
+NI about
+neighboring attacks and coupling deviations are taken into account. While this
+
+Springer Nature 2021 LATEX template
+14
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+Algorithm 2 Adaptively robust distributed MPC
+Input: local dynamic model for each subsystem I ∈ D,
+initial contracts �
+Zl,[0]
+I
+for all I, l, e.g., �
+Zl,[0]
+I
+= {hI(x0
+I)},
+finite parameter scenario sets �
+Wl,[k]
+I
+for all l, k
+1: set �
+Al,[0]
+I
+:= {} for all I, l
+2: for time step k do
+3:
+for I ∈ D do
+4:
+build scenario tree by branching on �
+Al,[k−1]
+I
+, �
+Zl,[k−1]
+NI
+, and �
+Wl,[k−1]
+I
+5:
+solve problem (9) to compute inputs ul
+I
+6:
+derive new contracts �
+Zl,[k]
+I
+▷ update contracts
+7:
+transmit �
+Zl,[k]
+I
+to neighbors
+8:
+end for
+9:
+apply first control input uk = (uk
+I)I∈D
+10:
+for I ∈ D do
+11:
+solve problem (5) to obtain a suspicion a∗,k
+I
+▷ local ADI
+12:
+update estimates µ[k]
+I , σ[k]
+I
+as in (10)
+13:
+adapt uncertainty set �
+Al,[k]
+I
+as in (11)
+▷ update attack scenarios
+14:
+end for
+15: end for
+is not reasonable if the neighbors and thus their transmitted sensitivities �Sa
+NI
+and �Sz
+NI are generally deemed untrustworthy, it is useful if the communication
+channel to the neighbors is considered secure, but the neighbors themselves do
+not apply ADI and therefore do not adapt their contracts to attacks.
+By enhancing distributed MPC with local attack identification in each
+subsystem, we obtain a distributed adaptively robust control framework, in
+which only locally available model knowledge and some information exchange
+among neighbors is involved. Unlike the related method introduced in [17],
+Algorithm 2 requires no central authority and, in particular, no confidential
+model knowledge is published globally. Such a procedure has the advantages
+that all local identification problems can be solved in parallel, that it can be
+employed even if the subsystems fail to agree on a central authority, and that
+no private model knowledge has to be shared with the entire network. Further-
+more, all distributed ADI approaches have in common that it is challenging to
+agree on system-wide countermeasures based on multiple, possibly contradic-
+tory local identification results. Our approach provides an answer to this issue
+as it transfers the insights from distributed ADI into local countermeasures by
+adjusting the local control inputs in a suitable robust way.
+5 Dynamic Model for Microgrids Under Attack
+Distributed microgrids that include local generation, demands, and often stor-
+age units, increase the security of supply within the microgrid area but create
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+15
+new challenges: Several optimal control tasks have to be addressed under the
+uncertainty of renewables and possibly even adversarial attacks, e.g., economic
+generator dispatch, efficient battery use, or optimal power import and export
+strategies to benefit from fluctuating energy prices [24, 25]. Therefore, we aim
+to apply the resilient control framework proposed in Section 4 to the task of
+microgrid control and derive a suitable dynamic model in this section.
+The main characteristics of the model are nonlinear battery dynam-
+ics, physical coupling of neighboring microgrids through dispatchable power
+exchange, and the threat of possible attacks. Each microgrid contains an aggre-
+gated load pl
+I ≤ 0 and a set of dispatchable generation units that generate a
+total power output pg
+I ≥ 0. How uncertain load and nondispatchable generation
+from renewable energy sources are modeled is discussed below. As illustrated
+in Figure 3, each microgrid is connected to the main grid, to or from which it
+can export or import power pm
+I ∈ R. While power import is modeled by posi-
+tive values pm
+I > 0, negative values pm
+I < 0 indicate power export to the main
+grid. In addition, power transfers are possible between two neighboring micro-
+grids I, L with L ∈ NI. The power that microgrid I provides to L is denoted
+as ptr
+IL and the resulting directed power flow from I to L is given as
+pflow
+IL := ptr
+IL − ptr
+LI.
+Finally, each microgrid has a storage unit that provides or consumes storage
+power pst
+I ∈ R and the state variable sI ∈ [0.0, 1.0] indicates its state of charge
+(SoC). Power values pst
+I > 0 indicate discharging and pst
+I < 0 charging. Unlike
+other works investigating economic dispatch problems in microgrid settings,
+for example Ananduta et al. in [15], we take into account that power cannot
+change instantaneously. Instead, the dynamic evolution of pg
+I, pm
+I , and ptr
+IL is
+controlled by inputs ug
+I, um
+I , and utr
+IL and behaves according to
+˙pg
+I = 1
+T g
+I
+(ug
+I + ag
+I − pg
+I) ,
+(12)
+˙pm
+I =
+1
+T m
+I
+(um
+I + am
+I − pm
+I ) ,
+(13)
+˙ptr
+IL =
+1
+T tr
+IL
+�
+utr
+IL + atr
+IL − ptr
+IL
+�
+.
+(14)
+The various delay parameters T g
+I , T m
+I , T tr
+IL ∈ R>0 depending on technical char-
+acteristics capture how quickly a change in the respective input affects the
+corresponding state. Compared to the generation delay T g
+I , typically smaller
+delay times T m
+I and T tr
+IL apply for power transfers with the main grid or neigh-
+boring microgrids. In line with the generic description of distributed systems
+under attack introduced in Section 2, we model attacks as additional, unknown
+inputs that impair the dynamic behavior of the microgrid systems as in (12)
+to (14). In each microgrid I ∈ D, we consider generator attacks ag
+I ∈ R, grid
+attacks am
+I ∈ R affecting the power exchange with the main grid, and transfer
+
+Springer Nature 2021 LATEX template
+16
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+pst
+I = -ΣpI
+pg
+I
+pl
+I
+pm
+I
+ptr
+IK
+ptr
+IL
+I
+L
+K
+zLI
+zKI
+Main grid
+Fig. 3: Schematic overview of the model for interconnected microgrids taken
+from [18, Fig. 1], showing the local model components for microgrid I. Apart
+from internal states, each microgrid only requires knowledge of its neighboring
+couplings (zLI)J∈NI. For power balance, storage units are used as a buffer.
+attacks atr
+IL ∈ R on power transfers to or from any neighbor L ∈ NI. While the
+inputs are computed by the local controller in I, the attack values are unknown
+to the control system. Thus, we deliberately make no difference in modeling
+attacks and renewable generation but consider both as uncertain influences
+resolved by the resilient control framework presented in Section 4.2. Similarly,
+uncertain load can be considered an attack al
+I modifying the load pl
+I = ul
+I that
+is modeled as a noncontrollable input with equal upper and lower bounds.
+The storage is used as a buffer providing the required power reserves at
+all times and thus assuring that the power balance in microgrid I is always
+satisfied, even when an attack occurs. Therefore, the storage power pst
+I is a
+dependent variable according to
+pst
+I = −pg
+I − pm
+I − pl
+I −
+�
+L∈NI
+�
+ptr
+LI − ptr
+IL
+�
+.
+It is important to distinguish that for microgrid I, the local state ptr
+IL can
+be controlled via utr
+IL as in (14), whereas the neighboring state ptr
+LI is neither
+controllable nor is its dynamic behavior known by microgrid I. The physical
+interconnection of neighboring microgrids is instead modeled by a coupling
+variable zLI = ptr
+LI and is treated locally as an uncertain parameter as we
+discussed in detail in Section 4.1. Figure 3 illustrates that the local knowledge
+is limited to local power variables and neighboring couplings.
+According to the storage power pst
+I , the storage is charged or discharged
+and the resulting change in the SoC sI is modeled as
+˙sI = bI
+�
+sI, pst
+I
+�
+with some function bI : [0.0, 1.0] × R → R modeling the battery dynamics.
+While a linear approximation of this charging behavior is usually sufficient in
+the middle range of [0.0, 1.0], it is not accurate for marginal values of the SoC
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+17
+which become extremely relevant in case of an attack. Following the line of
+[26, 27], the dynamics of the SoC are given as
+˙sI = − Ist
+I
+Qst
+I
+,
+(15)
+with Qst
+I denoting the maximum capacity of the battery and Ist
+I
+being the
+battery current. Denoting the battery voltage by U st
+I , the storage power pst
+I
+and the voltage U st
+I are given as
+pst
+I = U st
+I Ist
+I
+and U st
+I = U OCV
+I
+(sI) + Rst
+I Ist
+I .
+(16)
+in line with [26]. The term U OCV
+I
+denotes the open circuit voltage (OCV), that
+depends on the SoC sI, and the second summand determining U st
+I
+models
+the ohmic effect with resistance Rst
+I . Rewriting (16) results in the following
+relation for the storage power pst
+I :
+pst
+I = U OCV
+I
+(sI)Ist
+I + Rst
+I
+�
+Ist
+I
+�2 .
+Solving this equation for Ist
+I , the battery current Ist
+I = nI (sI, pst
+I ) is obtained
+from sI and pst
+I for some nonlinear function nI : [0.0, 1.0] × R → R. Together
+with (15), this results in a nonlinear function
+bI(sI, pst
+I ) := −nI(sI, pst
+I )
+Qst
+I
+that describes the dynamic behavior of the battery.
+It remains open to specify the open circuit voltage U OCV
+I
+(sI) using the
+model in [27], that is accurate also for low and high SOCs: With parameters
+αI, βI, γI, δI, µI, and νI depending on the type of battery, the OCV is given by
+U OCV
+I
+(sI) := αI + βI(−ln(sI))µI + γIsI + δIeνI(sI−1).
+(17)
+Bringing all of the above together, we have characterized a distributed
+dynamic system of interconnected microgrids, which results in a model of the
+form as in (2) when discretizing. Each microgrid is described by a local state
+xI =
+�
+sI, pg
+I, pm
+I , ptr
+I
+�⊤ ∈ R3+|NI|
+(18)
+with ptr
+I := (ptr
+IL)L∈NI and controlled by a local input
+uI =
+�
+ug
+I, um
+I , utr
+I
+�⊤ ∈ R2+|NI|,
+(19)
+that may be disturbed by an attack input
+aI =
+�
+ag
+I, am
+I , atr
+I
+�⊤ ∈ R2+|NI|
+(20)
+
+Springer Nature 2021 LATEX template
+18
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+with utr
+I := (utr
+IL)L∈NI and atr
+I := (atr
+IL)L∈NI. Power transfers to other micro-
+grids physically couple neighboring microgrids to each other, which is modeled
+by local coupling variables
+zI =
+�
+ptr
+IL
+�⊤
+L∈NI
+with zNI =
+�
+ptr
+LI
+�⊤
+L∈NI .
+(21)
+Each microgrid I ∈ D is operated locally to meet the respective load pl
+I at
+the lowest possible cost according to some objective function JI : R>0 → R,
+which specifies the costs incurred during some time window [0, T] of length
+T ∈ R>0 and is defined as
+JI(T) :=
+� T
+0
+qI
+�
+pg
+I, ptr
+I , pst
+I
+�
++ ℓI
+�
+pflow
+I
+, pm
+I
+�
+dt + mI (sI (T)) .
+(22)
+It consists of quadratic stage costs qI, piecewise linear stage costs ℓI, and
+terminal costs mI. The quadratic costs qI : R≥0 × RnzI × R → R≥0 with cost
+parameters Cg
+I , Ctr
+I , Cst
+I ∈ R≥0 are given as
+qI
+�
+pg
+I, ptr
+I , pst
+I
+� := Cg
+I (pg
+I)2 +
+�
+L∈NI
+Ctr
+I
+�
+ptr
+IL
+�2 + Cst
+I
+�
+pst
+I
+�2 .
+They capture the per-unit costs of using the units for power generation, power
+transfers to neighbors, and the respective storage operations. In contrast, the
+piecewise linear costs ℓI model the economic profit or loss from selling or
+buying energy in trade with neighbors or the main grid. Defining the positive
+and negative part functions
+(v)+ :=
+�
+0 if v < 0,
+v if v ≥ 0,
+and (v)− :=
+�
+v if v < 0,
+0 if v ≥ 0,
+the piecewise linear cost function ℓI : RnzI × R → R is given as
+ℓI
+�
+pflow
+I
+, pm
+I
+� :=
+�
+L∈NI
+Cflow,ex
+LI
+�
+pflow
+LI
+�
+− +
+�
+L∈NI
+Cflow,im
+LI
+�
+pflow
+LI
+�
++
++ Cm,ex
+I
+(pm
+I )− + Cm,im
+I
+(pm
+I )+
+for each microgrid I ∈ D, with local export and import per-unit prices Cflow,ex
+LI
+,
+Cflow,im
+LI
+, Cm,ex
+I
+, Cm,im
+I
+∈ R≥0, which may fluctuate throughout the day. In
+the numerical example in Section 6, we will consider import prices that are
+considerably higher than the export prices and thus focus on small producers,
+for which in practice it is often more profitable to generate power for their
+own demand than to buy electricity from the main grid, and for which power
+exports to the grid are only worthwhile at times of high demand. The terminal
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+19
+costs mI : [0.0, 1.0] → R≥0 account for degradation costs of the battery as
+mI(sI) := Cdis
+I
+(sI(0) − sI(T))+ Qst
+I .
+If the state of charge sI(T) at the end of the considered horizon is smaller than
+sI(0) at the beginning, each unit of power discharge is penalized by some cost
+Cdis
+I
+∈ R≥0.
+6 Numerical Experiments with Microgrids
+Under Attack
+In this section, we present a numerical case study to analyze the performance
+of adaptively robust DMPC from Section 4 in the context of interconnected
+microgrids under attack using the model from Section 5. In contrast to our
+earlier work [17], we apply distributed ADI based on the local identification
+problem (5). In the experiments, we address the question of how to achieve
+an economic operation of microgrids at minimum costs despite uncertainties.
+Whether these emerge in form of disturbances with rather small impact, fluc-
+tuating generation from renewables, or malicious attacks; all represent critical
+yet all the more relevant threats to energy supply.
+To this end, we consider three microgrids I, II, and III with renewable gen-
+eration that are each connected to the main grid and the other two microgrids
+as in Figure 3. The initial values and bounds for all variables of the microgrid
+model are given in Table 1 and the parameters are chosen as in Table 2, using
+those for lithium-titanate (Li4Ti5O12) batteries from [27] in (17).
+For a timespan of two days, robust NMPC is applied locally with step size
+∆t = 0.25 h by each microgrid. At time t ∈ [0.0, 48.0] h, the local cost func-
+tion JI in (22) takes into account the upcoming time window [t, t + Np] with
+prediction horizon Np = 6.0 h and uses the cost parameters from Table 2. The
+values Cm,im
+I
+and Cm,ex
+I
+, that describe the cost or revenue of power imports
+from or exports to the main grid, vary in the course of the day. In our example,
+we focus on microgrids that represent small local prosumers and use the fol-
+lowing fictitious values for all microgrids, which are based on real prices on the
+Table 1: This table lists lower and upper bounds as well as initial values at
+time t = 0 for all state and input variables of the microgrid model. For the state
+of charge, three distinct initial values sI(0) for the three microgrids I, II, and
+III are given. In all other cases, the indicated values apply for all subsystems.
+Variable
+Lower Bound
+Upper Bound
+Initial Value
+Unit
+sI
+0.0
+0.1
+0.9, 0.5, 0.6
+-
+pg
+I
+0.0
+1000.0
+0.0
+kW
+pm
+I
+-1000.0
+2000.0
+0.0
+kW
+ptr
+IL
+-100.0
+100.0
+0.0
+kW
+ug
+I
+0.0
+1000.0
+-
+kW
+um
+I
+-1000.0
+2000.0
+-
+kW
+utr
+IL
+-100.0
+100.0
+-
+kW
+
+Springer Nature 2021 LATEX template
+20
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+Table 2: This table lists all model and cost parameters that are used in
+the numerical experiments presented in this section. All values apply to all
+subsystems I ∈ {I, II, III}, except for Qst
+I , Rst
+I , and Cg
+I , where individual values
+for the respective subsystems are specified.
+(a) Model Parameters
+Param.
+Value
+Unit
+pl
+I
+-2.0
+kW
+T g
+I
+0.1
+h
+T m
+I
+0.001
+h
+T tr
+IL
+0.001
+h
+Qst
+I
+100, 200, 100
+kAh
+Rst
+I
+1.5, 2.0, 3.0
+mΩ
+(b) OCV Parameters
+Param.
+Value
+Unit
+αI
+2.23
+V
+βI
+-0.001
+V
+γI
+-0.35
+V
+δI
+0.6851
+V
+µI
+3.0
+-
+νI
+1.6
+-
+(c) Cost Parameters
+Param.
+Value
+Cg
+I
+0.2, 3.0, 2.0
+Ctr
+I
+4.0
+Cst
+I
+1.0
+Cdis
+I
+2000
+Cflow,im
+IL
+4.0
+Cflow,ex
+IL
+0.04
+German electricity market in 2021 [28] and reflect typical market fluctuations
+with rising prices in the morning and evening hours:
+Cm,im
+I
+(t) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+275 if (t mod 24 h) ∈ [15, 20) h,
+200 if (t mod 24 h) ∈ [6, 9) ∪ [20, 22) h,
+150 if (t mod 24 h) ∈ [9, 15) ∪ [22, 24) h,
+100 otherwise,
+Cm,ex
+I
+(t) =
+�
+�
+�
+�
+�
+15 if (t mod 24 h) ∈ [15, 20) h,
+10 if (t mod 24 h) ∈ [6, 9) ∪ [20, 22) h,
+0 otherwise.
+Here, mod is the modulo operator and (t mod 24 h) denotes the time of day.
+To achieve a resilient operation, the system is controlled using the adap-
+tively robust distributed NMPC scheme from Section 4.2. Based on the local
+control problem (9), at each sampling time k every microgrid computes con-
+tracts
+�
+X l,[k]
+I
+to confine the behavior of its future coupling values zl
+I for
+l ∈ {k, . . . , k + Np − 1} and shares them with its neighbors. In contrast to the
+experiments in [17], which involve a centralized ADI method, each microgrid
+consults locally identified solutions a∗,k
+I
+of problem (5) to update its estimates
+µ[k]
+I
+and σ[k]
+I
+of the expected value and standard deviation of the unknown
+random attack aI as in (10). In our numerical experiments, the nonlinear
+identification problem (5) is solved to an accuracy of εI = 10−3 using the
+interior-point solver Ipopt [29]. The states xI are assumed to be only partially
+observable with linear output function cI : XI → YI that is defined as
+cI (xI) := diag(1, 1, 1, 0, 0)xI.
+This means that for each microgrid I, the outputs yI = (sI, pg
+I, pm
+I )⊤ are con-
+sidered by the local identification process, but not the transfer variables ptr
+IL
+for all L ∈ NI. Based on the suspected attacks a∗,k
+I
+and the derived estimates
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+21
+                       
+                       
+     
+     
+      
+     
+     
+sI robust
+sI non-robust
+SoC in %
+100
+96
+92
+88
+(a) State of Charge
+                       
+                       
+     
+     
+      
+     
+     
+Generation in kW
+25
+20
+15
+10
+5
+0
+ug
+I
+pg
+I
+(b) Power Generation
+                       
+                       
+     
+     
+      
+     
+     
+Time in hours
+0.0
+12.0
+24.0
+36.0
+48.0
+Imports / exports in kW
+0
+-5
+-10
+-15
+-20
+-25
+-30
+pm
+I
+(c) Power exchange with main grid
+                       
+                       
+     
+     
+      
+     
+     
+Time in hours
+0.0
+12.0
+24.0
+36.0
+48.0
+Transfers in kW
+2.0
+1.5
+1.0
+0.5
+0.0
+ptr
+I,II
+ptr
+I,III
+(d) Power exchange with neighbors
+Fig. 4: Selected state and input trajectories for microgrid I, showing all powers
+in kW. The microgrid is exposed to a generator attack, causing the generation
+pg
+I to be considerably larger than planned by ug
+I . The different SoC trajectories,
+computed by adaptively robust versus nonrobust NMPC, show the benefit of
+the proposed resilient control framework.
+µ[k]
+I
+and σ[k]
+I , the uncertainty sets �
+Al,[k]
+I
+are approximated as in (11). The local
+control problem (9) is repeatedly adapted to new contracts and identification
+results that become available. As a consequence, the inputs ul
+I computed at
+time k+1 for l ∈ {k+1, . . . , k+Np} are robust toward deviations in neighboring
+couplings within �
+Zl,[k]
+NI
+and identified attacks in �
+Al,[k]
+I
+.
+We examine the behavior of the system, controlled with Algorithm 2, in
+two attack scenarios. For comparison, we repeat each experiment with nonro-
+bust DMPC, where neither contracts are exchanged nor attack identification
+is considered. First, we assume that all generation units are dispatchable and
+a constant attack ag
+I = 10.0 kW disrupts the generator dynamics in micro-
+grid I according to (12). The attacker is active over the entire time window
+[0.0, 48.0] h and causes a severe deviation of the generated power pg
+I in micro-
+grid I from the control input ug
+I as Figure 4 reveals. The distributed ADI
+method based on the local identification problem (5) successfully identifies
+the unknown attack input with very high precision in every time step as
+pointed out by Figure 5, which shows the mean of the suspected attack values
+ag,∗
+I
+≈ 9.9989 kW at all times. This allows the local robust NMPC scheme to
+adapts its prediction very accurately and adjust the control inputs accordingly.
+As a result, the microgrid takes advantage of the additional power generation
+
+Springer Nature 2021 LATEX template
+22
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+                                        
+                                        
+Actual attack ag
+I
+Identified mean µ[k]
+I
+Time in hours
+0.0
+12.0
+24.0
+36.0
+48.0
+Attack ag
+I in kW
+10.015
+10.010
+10.005
+10.000
+9.995
+Fig. 5: Actual attack value ag
+I and average identified value µ[k]
+I
+in the first
+attack scenario examined, in which only dispatchable generation units are in
+use and microgrid I is exposed to a generator attack.
+by charging the battery and exporting the power to the main grid during times
+with high profit. In the solution computed with nonrobust NMPC, on the con-
+trary, the battery reaches and violates its maximum state of charge of 1.0 after
+about 5.0 h as the red SoC trajectory in Figure 4a reveals. Due to bound vio-
+lations, the nonrobust scheme fails in 171 of 192 time steps when more power
+than planned is generated and the storage is charged to maintain power bal-
+ance. Since SoC values larger than 1.0 are physically invalid, the next MPC
+step in our study continues at sI = 1.0.
+It should be noted that power balance can be ensured in other ways than
+using the storage as a buffer. For instance, if power imports from and exports
+to the main grid are allowed at all times, using the grid as a buffer would not
+lead to bound violations as above. However, this can cause very high costs, for
+example, if electricity has to be imported in the evening at expensive prices. In
+contrast, the battery allows power to be stored until exports to the main grid
+become profitable. Indeed, over the entire period of two days, the adaptively
+robust NMPC scheme achieves total costs of −5.2·103 in microgrid I and thus
+makes profit despite the attack. On the contrary, nonrobust NMPC causes
+total local costs of 2.3 · 104, which is orders of magnitudes larger. Considering
+that we aim for a strategy to increase the resilience of the system, which takes
+into account not only robustness but also performance in terms of induced
+costs, the battery as a buffer is therefore a reasonable choice that enables and
+favors high resilience.
+In the second experiment, we consider a modified generator attack
+ag
+I = 10.0 kW + rg
+I , where rg
+I ∼ N(0.0, 8.0) kW represents the uncertainty in
+renewable generation and is randomly drawn from a normal distribution with
+mean 0.0 kW and standard deviation 8.0 kW, independently at each time step.
+Together, the malicious attack of 10.0 kW and the renewable fluctuations rg
+I
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+23
+                                        
+                                        
+                                        
+Actual attack ag
+I
+Identified mean µ[k]
+I
+Sample std. dev. σ[k]
+I
+Time in hours
+0.0
+12.0
+24.0
+36.0
+48.0
+Disturbance ag
+I in kW
+40.0
+30.0
+20.0
+10.0
+0.0
+-10.0
+Fig. 6: Course of the mean µ[k]
+I
+of identified values a∗,k
+I
+over time, with sample
+standard deviation σ[k]
+I . The actual disturbance ag,k
+I
+at each time k is shown
+in orange. The figure is taken from [18, Fig. 3].
+may cause more power than planned to be generated (i. e., ag
+I > 0) or less (i. e.,
+ag
+I < 0), but are chosen such that the total generator input ug
+I + ag
+I is nonneg-
+ative. Due to the fluctuating generation, the actual value ag
+I of the unknown
+disturbance in the generator dynamics ranges from −11.3 kW to 42.9 kW as
+can be seen in Figure 6. For the examined generator with parameters as in
+Table 2, this is a very broad range, which also becomes clear in comparison
+with Figure 4b. As an apparent consequence of the continually changing val-
+ues, the local identification problem (5) yields a different suspicion ag,∗
+I
+in each
+time step. Nevertheless, Figure 6 shows that the mean µ[k]
+I
+of identified values
+quickly settles at about 10.0 kW, which underlines that the distributed ADI
+method is able to cope also with highly fluctuating and widely dispersed dis-
+turbances, since a new optimization problem is solved at each time step. This
+proves once again the great potential of the proposed class of optimization-
+based ADI methods and emphasizes that they are not tailored to a specific
+type of attack, but are also very well suited for challenging scenarios where
+attacks and other sources of significant uncertainty congregate.
+The sample standard deviation σ[k]
+I
+is considerably larger than before and
+the three scenarios µ[k]
+I , µ[k]
+I
++ σ[k]
+I , and µ[k]
+I
+− σ[k]
+I
+are further apart than in
+the first experiment. Figure 7 shows the obtained solution for the attacked
+microgrid I. While adaptively robust DMPC achieves total local costs of 3.1·103
+in microgrid I, the nonrobust approach causes more than ten times higher total
+costs of 3.2·104. Once again, classical nonrobust MPC proves to be unsuitable
+
+Springer Nature 2021 LATEX template
+24
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+                       
+                       
+     
+     
+      
+     
+     
+sI robust
+sI non-robust
+SoC in %
+100
+95
+90
+85
+80
+(a) State of Charge
+                       
+                       
+     
+     
+      
+     
+     
+Generation in kW
+60
+40
+20
+0
+ug
+I
+pg
+I
+(b) Power Generation
+                       
+                       
+     
+     
+      
+     
+     
+Time in hours
+0.0
+12.0
+24.0
+36.0
+48.0
+Imports / exports in kW
+0
+-10
+-20
+-30
+-40
+pm
+I
+(c) Power exchange with main grid
+                       
+                       
+     
+     
+      
+     
+     
+Time in hours
+0.0
+12.0
+24.0
+36.0
+48.0
+Transfers in kW
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+ptr
+I,II
+ptr
+I,III
+(d) Power exchange with neighbors
+Fig. 7: States and inputs in microgrid I, which now contains renewable
+generation as another source of uncertainty in addition to the generator attack.
+to control the disturbed system as it computes a solution that violates the
+upper bound of the state of charge in 113 of 192 time steps.
+At this point, we would like to point out that the adaptively robust DMPC
+scheme is not guaranteed to yield admissible trajectories in all cases. In fact,
+proving rigorous guarantees of this kind is challenging for nonlinear dynam-
+ics. Moreover, in contrast to the multi-stage approach [30], adaptively robust
+NMPC lacks the recursive feasibility property when the attack uncertainty sets
+Al,[k]
+I
+are adjusted to sudden attacks. Furthermore, Figure 6 illustrates that
+in our second attack scenario involving uncertain renewable generation, even
+disturbances ag
+I occur that are not within the interval [µ[k]
+I
+− σ[k]
+I , µ[k]
+I
++ σ[k]
+I ].
+Despite these unforeseen disruptions and the lack of theoretical guarantees,
+however, all state bounds are satisfied and the solution in Figure 7 is not overly
+conservative judging from the fact that considerably lower costs are obtained
+than with nonrobust DMPC. This underlines that adaptively robust NMPC,
+using ADI results as estimates for an unknown attack, is a very powerful tool
+even under challenging circumstances with broadly dispersed disturbances.
+7 Conclusion and Future Directions
+We introduced a comprehensive distributed MPC framework for nonlinear
+control systems under attack, which is based on local multi-stage control and
+novel distributed attack identification methods in each subsystem. To enable
+
+Springer Nature 2021 LATEX template
+Resilient MPC of Distributed Systems Under Attack Using Local ADI
+25
+the system to respond autonomously and robustly to identified perturbations,
+each control scheme represents the uncertain influence of neighboring cou-
+plings and attack inputs by scenario sets that are continuously updated based
+on newly gained knowledge. For this purpose, each subsystem applies local
+attack identification and repeatedly transmits new contract information to its
+neighbors. Using the example of microgrids interconnected by power transfers,
+the methodology was demonstrated to robustly control a distributed system
+and achieve constraint satisfaction at all times despite unknown attacks and
+uncertain renewable generation.
+We have identified two promising directions with great potential for future
+research. The first would be to derive theoretical conditions under which Algo-
+rithm 1 can be rigorously proven to successfully identify the correct inputs,
+similar to the guarantees for our centralized ADI method [16]. While some
+ideas from [16] can be transferred with few changes, further required theoreti-
+cal arguments could be based on the research results on nonlinear compressed
+sensing. For example, in [22] the restricted isometry property from [20], a cen-
+tral component of linear compressed sensing, is generalized and the iterative
+hard thresholding algorithm involving a form of gradient projection is extended
+to nonlinear systems. Furthermore, in [23] two coordinate descent methods
+are introduced that build upon the simplex algorithm for linear programming
+and are of a greedy type in the sense that they add nonzero variables one by
+one. When suitable success guarantees for the new distributed ADI approaches
+provably hold, a combination with the robustness and stability analysis of
+multi-stage NMPC and contract-based DMPC described in [3, 30, 31] could be
+the next step to strengthen the excellent numerical performance of adaptively
+robust DMPC by theoretical arguments.
+The second research direction consists in investigating a hierarchical com-
+bination of several ADI approaches that complement each other and provide
+system operators with different options suiting their needs. There is, on the
+one hand, the centralized ADI method from [16], which is based on an approx-
+imation of the dynamics and provides quick insights into the network-wide
+attack situation, but requires all subsystems to make specific sensitivity infor-
+mation publicly available and agree on a central instance to solve the global
+identification problem. On the other hand, there are distributed ADI methods
+like Algorithm 1 involving problems (5) and (8), which use local models to
+analyze possible attacks on one subsystems or its neighborhood locally. Sev-
+eral gradations or variants of these approaches may be applied, depending on
+the available model knowledge and the willingness of individual subsystems to
+cooperate or agree on a common decision instance.
+8 Statement on Conflict of Interests
+On behalf of all authors, the corresponding author states that there is no
+conflict of interest.
+
+Springer Nature 2021 LATEX template
+26
+REFERENCES
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+

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+QoS Based Contract Design for Profit Maximization
+in IoT-Enabled Data Markets
+Juntao Chen, Member, IEEE, Junaid Farooq, Member, IEEE and Quanyan Zhu, Senior Member, IEEE
+Abstract—The massive deployment of Internet of Things (IoT)
+devices, including sensors and actuators, is ushering in smart and
+connected communities of the future. The massive deployment of
+Internet of Things (IoT) devices, including sensors and actuators,
+is ushering in smart and connected communities of the future.
+The availability of real-time and high-quality sensor data is
+crucial for various IoT applications, particularly in healthcare,
+energy, transportation, etc. However, data collection may have
+to be outsourced to external service providers (SPs) due to
+cost considerations or lack of specialized equipment. Hence,
+the data market plays a critical role in such scenarios where
+SPs have different quality levels of available data, and IoT
+users have different application-specific data needs. The pairing
+between data available to the SP and users in the data market
+requires an effective mechanism design that considers the SPs’
+profitability and the quality-of-service (QoS) needs of the users.
+We develop a generic framework to analyze and enable such
+interactions efficiently, leveraging tools from contract theory and
+mechanism design theory. It can enable and empower emerging
+data sharing paradigms such as Sensing-as-a-Service (SaaS). The
+contract design creates a pricing structure for on-demand sensing
+data for IoT users. By considering a continuum of user types,
+we capture a diverse range of application requirements and
+propose optimal pricing and allocation rules that ensure QoS
+provisioning and maximum profitability for the SP. Furthermore,
+we provide analytical solutions for fixed distributions of user
+types to analyze the developed approach. For comparison, we
+consider the benchmark case assuming complete information of
+the user types and obtain optimal contract solutions. Finally, a
+case study based on the example of virtual reality application
+delivered using unmanned aerial vehicles (UAVs) is presented
+to demonstrate the efficacy of the proposed contract design
+framework.
+Index Terms—Contract design, data pricing, Internet of things,
+Maximum principle, quality-of-service, sensing-as-a-service.
+I. INTRODUCTION
+The Internet of things (IoT) applications rely heavily on
+sensed data from a multitude of sources resulting in power-
+ful and intelligent applications based on sensor fusion and
+machine learning. For instance, smart and connected commu-
+nities, industrial automation, smart grid all rely on reliable
+and high quality data for automated decision-making [1]. To
+This work was supported in part by the National Science Foundation (NSF)
+under Grants ECCS-1847056, CNS-2027884, and BCS-2122060.
+Juntao Chen is with the Department of Computer and Information
+Sciences,
+Fordham
+University,
+New
+York,
+NY
+10023
+USA.
+E-mail:
+[email protected].
+Junaid Farooq is with the Department of Electrical & Computer Engineer-
+ing, College of Engineering and Computer Science, University of Michigan-
+Dearborn, Dearborn, MI 48128 USA. E-mail: [email protected].
+Quanyan Zhu is with the Department of Electrical and Computer Engi-
+neering, Tandon School of Engineering, New York University, Brooklyn, NY,
+11201 USA. E-mail: [email protected].
+UAVs
+VR users
+VR SP
+VR 
+services
+Service 
+fee
+Sensing 
+data
+Fig. 1.
+In the UAV-enabled VR applications, the UAVs capture views
+of the areas of interest. The collected data are aggregated in the cloud,
+which is managed by the VR SP, and then sent to the remote users. The
+real-time 3D information delivery is useful in applications such as remote
+monitoring, navigation, and entertainment. Based on the application, VR users
+have different QoS requirements and pay different service fees.
+fulfil the data needs of intelligence-based IoT applications,
+the sensing and data acquisition tasks can be outsourced to
+professional service providers (SPs) in the data market [2].
+It results in cost effective data collection for IoT applica-
+tions, wider choice of sensing data, and on-demand service
+delivery to users. For example, in an intelligent transportation
+network, vehicles can choose the services to communicate
+with roadside infrastructures that belong to sensing SP for
+exchanging various types of data related to applications such
+as GPS navigation, parking, and highway tolls inquiries. etc.
+Another potential scenario is UAV-enabled virtual reality (VR)
+experiences [3]. As shown in Fig. 1, the UAVs managed by
+the SP capture 3D images of areas that users are interested
+in, and send them to the remote users via cloud servers and
+communication networks. These images can be of varying
+quality and resolution suited for a range of different user
+types. Therefore, the service interactions between the users
+and the sensing SP requires a formal contract design, in
+which IoT users make subscription contracts with the SP to
+obtain (real-time) sensor data according to specific mission
+requirements [4].
+Depending on the particular application, IoT users have
+different requirements on the quality of data provided by the
+sensing SP. Note that provisioning of high-quality sensing
+data demands high-level of investment in terms of equipment
+deployment, maintenance, technical support, and data process-
+ing from the SP. In the UAV-enabled VR, users may require
+different levels of quality-of-service (QoS) in terms of the
+transmission delay and resolution of the images. Therefore,
+users with different QoS needs can be classified into different
+arXiv:2301.04691v1  [eess.SY]  11 Jan 2023
+
+HAKIOABtypes1. The sensing SP aims to maximize its revenue and
+minimize the service costs jointly by delivering on-demand
+sensing services. In contrast, the user’s goal is to choose
+a service that maximizes its utility. Therefore, there is a
+need to design efficient contracting strategies between the SP
+and the users so that sensing technologies can be effectively
+monetized. In the proposed contract design framework, the SP
+needs to design a menu of contracts that specify the sensing
+price and the QoS offered to each type of user. The optimal
+contracts yield a matching between the available sensing data
+and users in the IoT ecosystem that is suitable for both SP
+and the users.
+Due to the large-population feature of users in the massive
+IoT [5], the SP may not be aware of the exact type of
+each user and may only have high level information on
+the distribution of user’s types (e.g., inferred from historical
+demand data)2. Thus, the challenge of contract design lies
+in the development of an incentive compatible and optimal
+mechanism for the sensing SP to maximize its payoff by
+serving IoT users inspite of the incomplete information. To
+overcome this obstacle, we propose a market-based pricing
+contract mechanism for the SaaS model that takes into account
+incentive compatibility and individual rationality of the users.
+Specifically, we consider a continuum of user types with a
+generic probability distribution and design optimal contracts
+leveraging the Pontryagin maximum principle [6].
+Under a wide class of probability distributions of user’s
+type, we obtain analytical expression of optimal contracts in
+which the pricing scheme and the QoS mapping are mono-
+tonically increasing with user’s types, creating a complete
+sensing service market with all possible QoS levels. When the
+probability density function of user’s type distribution has a
+large or sudden decrease around some points, then nondis-
+criminative pricing phenomenon occurs, which reduces the
+diversity of service provisions to the IoT users. Specifically,
+some users choose the same service contract in spite of their
+heterogeneous types. In addition, nondiscriminative pricing for
+all customers can occur when the user’s types are nested in
+the lower regime. Hence, in this scenario, the SP should target
+at the majority in the market to optimize the revenue. For
+comparison, we study the optimal contracts under complete
+information and characterize the solution differences.
+We illustrate the optimal SaaS mechanism design principles
+with an application to the UAV-enabled VR. Simulation results
+show that the SP earns more profit by serving users with rela-
+tively stringent service requirements (higher types). However,
+since the users of lower types constitute most of the market,
+the SP gains a large proportion of revenue from serving low
+type users even though their unit benefit is smaller.
+The main contributions of this paper are summarized as
+follows:
+1) We propose a two-sided market-based SaaS contract
+design for QoS driven data trading between the service
+1The user types can also be interpreted as the importance of tasks to the
+users respectively, ranging from non mission-critical to mission-critical ones.
+2This asymmetric information assumption also aligns with the fact that the
+users aim to preserve privacy of their true types.
+provider and users in the IoT ecosystem under asym-
+metric information.
+2) We characterize the solutions of optimal contracts for
+arbitrary distributions of user types, that yield the best
+matching between the sensing services and the users
+leveraging the Pontryagin maximum principle.
+3) We show that under the efficient data pricing mecha-
+nism, the optimal contracts either capture the diversity
+of user types (discriminative pricing) or focus on the
+majority of user types (nondiscriminative pricing) de-
+pending on the users’ preferences.
+4) We provide an illustrative example of UAV-enabled VR
+application to validate and test our proposed contract
+design. We further provide a comparison between the
+hidden and full information scenarios in terms of the
+payoff of the SP.
+A. Related Work
+Contract design [7] has typically been used in operations
+research with applications to retail, financial markets [8],
+insurances [9], supply chains [10], etc. With the emergence
+of IoT and the data markets [11], new service models such
+as the SaaS are being developed enabling new possibilities
+such as resource trading [12], [13], opportunistic IoT [14],
+task offloading and outsourcing [15], and performance oriented
+resource provisioning [16], [17]. Therefore, there is a need for
+developing effective contracts [18] and pricing schemes [19],
+[20] that incentivize the interactions between users and service
+providers of data in the IoT ecosystem. The data markets
+and contract solutions can be implemented using blockchain
+infrastructure over IoT networks [21]–[23].
+A variety of literature is available on using contract theory
+for incentive mechanism design in wireless communication
+systems [24], tailored for scenarios such as traffic offloading
+[25], [26], relay selection [27], spectrum trading [28], [29],
+etc. In [30], the authors have studied the resource trading pro-
+cess between a mobile virtual wireless network operator and
+infrastructure providers using a contract. Similar approaches
+have also been used to facilitate Wi-Fi sharing in crowdsourced
+wireless community networks [31]. Incentive mechanism de-
+sign has also been received a lot of attention in the next-
+generation crowdsensing applications. For example, a two-
+stage Stackelberg game approach has been proposed in [32]
+to design incentive mechanism for the crowdsensing service
+provider by capturing the participation level of the mobile
+users. In [33], the authors have investigated the sequential dy-
+namic pricing scheme of a monopoly mobile network operator
+in the social data market by considering the congestion effects
+in wireless networks. In [34], a distributed computing approach
+is used in crowdsourcing using contracts by focusing on
+designing a reward-based collaboration mechanism. Contract
+theory is also leveraged to price the sponsored content in mo-
+bile service [35], where the authors developed a hierarchical
+game framework to capture the service relationships between
+the network operator acting as the leader and the content
+provider and the end users acting as followers.
+Our work focuses on establishing a sensing data trading
+platform enabled by the IoT by considering the user’s ratio-
+
+nality and market reputation in a holistic manner. Different
+from [36] where the authors have focused on designing a
+pricing mechanism for data delivery in massive IoT from
+a routing perspective, we address the data pricing problem
+based on a contract-theoretic approach. Regarding SaaS in
+the IoT, [37] has established a public sensing framework
+for service-based applications in smart cities where the data
+is provided by a cloud platform. The authors in [38] have
+investigated smart phone-based crowdsensing to enhance the
+public safety via the collected sensing data. In this paper, we
+use an analytical approach to create an implementable policy
+framework, focusing on a large-population regime through
+contract design, which facilitates the realisation of the SaaS
+paradigm.
+We highlight several differences of this work with the
+literature that uses contracts in various service provisioning
+applications related to IoT and (wireless) communications.
+Different from the majority of works (e.g., [25], [26], [28]–
+[35]) that have considered finite number of user ‘types’ in
+the contract formulation, our framework focuses on a large-
+population regime of IoT users and uses a density function
+to describe the heterogeneous types of users. The second
+difference is that our framework considers the reputation of
+service provisioning through an average QoS constraint. This
+constraint implicitly improves the inclusion of distinct types
+of users in the service market. The third difference is on the
+solution approach used. Instead of solving the problem from an
+classical optimization angle, this work addresses the problem
+from an optimal control perspective.
+B. Organization of the Paper
+The rest of the paper is organized as follows. Section II
+introduces the SaaS framework and formulates the contracting
+problem. Contract analysis under a class of user’s type distri-
+butions is presented in Section III. We provide the detailed
+optimal contract solutions for two special cases in Section
+IV. Section V investigates the contract design under complete
+information. Extensions of the contract design to general user’s
+type distributions are presented in Section VI. Section VII
+illustrates the obtained results with an application to UAV-
+based VR, and Section VIII concludes the paper.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+We consider a pool of IoT users with varying QoS require-
+ments, that are connected to an SP for obtaining sensing data.
+We assume that the SP has similar sensing data available in
+a variety of different quality levels. For instance, the same
+video data can be available in many different pixel resolutions.
+Each user obtains a particular quality of data from the SP for
+its specific mission needs. Depending on the application and
+quality of data required, the IoT users can be characterized by
+their ‘type’, denoted by δ. In the following subsections, we
+provide a description of the different model parameters and
+an analytical formulation of the optimal contract between an
+SP and IoT users.
+ 
+55%
+28%
+12%
+3%
+1%
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+Less than $250
+$250 to $400
+$400 to $600
+$600 to $1000
+More than $1000
+Percentage of customers
+Spending preferences
+Types of Customers in VR
+Data Points
+Exponential Fitting
+Fig. 2. Customers’ spending preferences in the VR headset. Note that a higher
+price of VR equipment can be interpreted as the customer preferring higher
+quality of VR experiences. Then, the data yields an empirically exponential
+distribution of the customers’ types in our contract design for VR services.
+A. User Type and Data Quality
+Considering a large number of users in a massive IoT
+setting, each user is characterized by its type δ ∈ ∆ := [δ, ¯δ],
+which is hidden to the SP, where δ ≥ 0 and ¯δ ≥ 0 denote the
+lower and upper bounds of the parameter, respectively. Here, δ
+signifies the importance level of user’s task depending on the
+application needs. Furthermore, considering a large number of
+possible user types, we assume a continuum of δ admitting a
+value from the set ∆. The incomplete information of the IoT
+users to SP implies that the SP does not know the individual
+attributes of the users. However, the SP may have a broad
+understanding of the probability distribution of the users. This
+preserves user’s privacy to a certain degree. Hence, instead
+of knowing the explicit information of δ, we assume that the
+sensing SP has knowledge only about the probability density
+function of the users’ type, denoted by f(δ).
+Example 1. Empirical Estimation of User Type Distribution
+To design practical contracts for VR services in the IoT, we
+plot the data of customers’ spending preferences on the VR
+equipment in Fig. 2. The data is adapted from [39]. Since a
+higher price of VR equipment generally yields a better quality
+of VR experience, the data can be used to approximate the
+distribution of customers’ types in our VR contract design.
+Fig. 2 depicts five levels of customers’ types. Without loss of
+generality, we can consider their types as type 0, type 1, type
+2, type 3, and type 4, respectively, from left to right. In the
+proposed SaaS framework, we consider an on-demand sensing
+service provision in a large-population regime. Thus, the cus-
+tomer’s type parameter is continuous over a bounded support.
+Motivated by Fig. 2, we consider the type parameter δ taking
+a value from the interval ∆ := [δ, ¯δ], where δ = 0, ¯δ = 4,
+and a larger δ indicates a higher requirement of VR data
+quality. This modeling is consistent with the statistics shown in
+Fig. 2. Furthermore, based on Fig. 2, δ empirically admits an
+exponential distribution. Using statistical inference techniques,
+we can obtain f(δ) = 0.952e−0.952δ, and F(δ) = 1−e−0.952δ.
+Note that these probability density and distribution functions
+are aligned with the market data in Fig. 2.
+
+The sensed data available to the SP is characterized by its
+QoS level, denoted by q ∈ R, and the corresponding price
+(payment by the user), denoted by p ∈ R. The QoS level can
+be related to a number of specific metrics, such as the pixel
+density, latency, and jitter in the transmission of sensing data,
+etc. Note that we consider a continuum of quality levels since
+a large number of different versions of data are assumed to be
+available to the SP. In general, we can consider a vectorized q,
+where each element denotes the quality of the corresponding
+metric. The set Q denotes the available QoS levels provided
+by the SP.
+B. QoS Provisioning and Profit of SP
+The service relationships between users and SP described
+above can be naturally captured by a contract-theoretic frame-
+work. Specifically, due to the asymmetric information induced
+by users’ hidden type, the SP needs to design a menu of
+contracts, i.e., {q(δ), p(δ)} and present it to the users. Each
+user will then choose one contract that maximizes its payoff.
+The payoff of the user with type δ, which claims to be of type
+δ′ (thus receiving contact {q(δ′), p(δ′)}) can be computed as
+V (δ, δ′) = Φ (δ, q(δ′)) − p(δ′),
+(1)
+where V : ∆ × ∆ → R, and Φ : ∆ × Q → R. Note that the
+function Φ is a measure of the utility of the user. A natural
+assumption of Φ is described as follows:
+Assumption 1. The function Φ is continuously differentiable
+and increasing in variables δ and q, i.e., ∂Φ(δ,q(δ))
+∂δ
+> 0 and
+∂Φ(δ,q(δ))
+∂q(δ)
+> 0. Also, it satisfies ∂Φ2(δ,q(δ))
+∂q(δ)∂δ
+> 0.
+Assumption 1 indicates that with a better QoS level, the
+payoff of user increases. Also, for a given QoS level, the
+users with a larger type parameter δ have a higher payoff
+since their tasks are more mission-critical. Furthermore, for a
+same amount enhancement of QoS level, the resulting payoff
+increases for higher type users exceeds the one associated with
+lower types users.
+The function describing the SP’s profit obtained by provid-
+ing a QoS level q to a user of type δ, is defined as
+U(δ) = p(δ) − C(q(δ)),
+(2)
+where C : Q → R+ is the cost of the SP for providing the
+sensor data. Then, the expected total payoff of the SP can be
+expressed as
+� ¯δ
+δ (p(δ) − C(q(δ)))f(δ)dδ, where f(δ) denotes
+the density of type δ users. We consider that f(δ) is strictly
+greater than 0, i.e., f(δ) > 0, ∀δ ∈ [δ, ¯δ], which holds in the
+case of massive IoT.
+C. Profit Maximizing Contract Problem
+Based on the direct revelation principle [40], it is sufficient
+for the SP to design/consider contracts in which the users can
+truthfully select the one that is consistent with their true types;
+in other words, the users will reveal their types in the selection
+and do not have incentives to misrepresent their true types.
+Hence, we characterize the incentive compatibility (IC) and
+individual rationality (IR) constraints of the users defined as
+follows.
+Definition 1 (Incentive Compatibility). A menu of contracts
+{q(δ), p(δ)}, ∀δ, designed by the sensing SP is incentive
+compatible if the user of type δ selects the contract (q(δ), p(δ))
+that maximizes its payoff, i.e.,
+Φ(δ, q(δ)) − p(δ) ≥ Φ(δ, q(δ′)) − p(δ′), ∀(δ, δ′) ∈ ∆2. (3)
+Definition 2 (Individual Rationality). The individual rational-
+ity constraint of each user is captured by
+Φ(δ, q(δ)) − p(δ) ≥ 0, ∀δ ∈ ∆.
+(4)
+To investigate the impact of average sensing QoS level on
+the optimal contracts, the SP has an additional constraint on
+the provided QoS to the IoT users as follows:
+� ¯δ
+δ
+q(δ)f(δ)dδ ≥ q,
+(5)
+where the positive constant q is the mean/average QoS. The
+constraint (5) can be interpreted as the reputation that the
+SP aims to build in the sensing service market. Note that
+the average QoS has been leveraged to guide the optimal
+decision-making in various applications in literature, such as
+bandwidth allocation in broadband service provisioning [41],
+admission control of services in edge computing [42], and
+transmit powers minimization in small cell base stations [43].
+The goal of the SP is to jointly determine the pricing scheme
+p(δ) and the corresponding service quality q(δ) that yields
+the best return. To this end, the SP is required to solve the
+following optimization problem:
+(OP) :
+max
+{q(δ),p(δ)}
+� ¯δ
+δ
+�
+p(δ) − C(q(δ))
+�
+f(δ)dδ
+s.t. Φ(δ, q(δ)) − p(δ) ≥ Φ(δ, q(δ′)) − p(δ′),
+∀(δ, δ′) ∈ ∆2, (IC)
+Φ(δ, q(δ)) − p(δ) ≥ 0, ∀δ ∈ ∆, (IR)
+� ¯δ
+δ
+q(δ)f(δ)dδ ≥ q. (Reputation)
+III. ANALYSIS AND DESIGN OF OPTIMAL CONTRACTS
+In this section, we first analyze the formulated problem (OP)
+in Section II. Then we design the optimal contracts for SaaS
+by using Pontryagin maximum principle [44].
+A. Problem Analysis
+To solve problem (OP), one challenge lies in the infinite
+number of IC and IR constraints in (3) and (4), respectively.
+To simplify (OP), we first present the following lemma.
+Lemma 1. Under the condition that p(δ) and q(δ) are
+differentiable, the set of IC constraints (3) is equivalent to
+the local incentive constraints
+dp(δ)
+dδ
+= ∂Φ(δ, q(δ))
+∂q(δ)
+dq(δ)
+dδ , ∀δ ∈ ∆,
+(6)
+
+and a monotonicity constraint
+dq(δ)
+dδ
+≥ 0.
+(7)
+Proof. See Appendix A.
+■
+To facilitate the optimal contract design in Section III-B, we
+specify some structures of the payoff, Φ and cost, C. The user
+with mission-critical tasks (higher δ) gains more by receiving
+better quality of sensor data (higher q). Therefore, a reasonable
+payoff function for type δ user can be chosen as follows.
+Assumption 2. The payoff function for type δ user is consid-
+ered as
+Φ(δ, q(δ)) = δq(δ).
+(8)
+Then, (6) can be simplified as
+dp(δ)
+dδ
+= δ dq(δ)
+dδ . Note that
+the payoff function (8) is not necessary linear. The only
+requirement of Φ is to satisfy Assumption 1. The analysis
+of optimal contract design in the following still holds for a
+general Φ. Similarly, to obtain analytical results of optimal
+contracts, one of the cost functions of sensing SP is chosen
+as follows.
+Assumption 3. The cost function of sensing SP is
+C(q(δ)) = σ (exp(aq(δ)) − 1) ,
+(9)
+where σ > 0 is a normalizing constant trading off between
+the sensing service costs and the revenue, and a > 0 is a
+sensitivity constant, indicating that the marginal cost of the
+sensor data is increasing with its quality.
+Corollary 1. Based on Lemma 1 and (8), the IC constraints
+in (OP) can be represented as
+dV
+dδ = q(δ),
+(10)
+together with the monotonicity constraint (7).
+Proof. Based on (1), we have dV
+dδ = dΦ
+dδ − dp(δ)
+dδ . Then, using
+dΦ
+dδ = δ dq(δ)
+dδ
++ q(δ) and dp(δ)
+dδ
+= δ dq(δ)
+dδ
+yield the result.
+■
+Note that Corollary 1 indicates that the payoff of the user
+is monotonically increasing with the type δ. Therefore, the IR
+constraint can be simpli��ed as Φ(δ, q(δ)) − p(δ) ≥ 0. Indeed,
+the IR constraint is binding under the optimal contracts for
+type δ users, i.e.,
+Φ(δ, q(δ)) − p(δ) = 0.
+(11)
+Otherwise, the sensing SP can earn more profits by increasing
+the price p(δ) for serving the type δ users.
+The reputation constraint (5) essentially divides the problem
+analysis in two regimes: whether the constraint is binding
+at the optimal solution or not. Denote by {p∗(δ), q∗(δ)} the
+optimal solution to (OP). When q is relatively large, then it is
+possible that
+� ¯δ
+δ q∗(δ)f(δ)dδ = q, since the SP has no incen-
+tive to provide a QoS q(δ) with q(δ) > q∗(δ) which decreases
+the objective value. The inequality
+� ¯δ
+δ q∗(δ)f(δ)dδ > q could
+happen when q is relatively small. Thus, there exists a thresh-
+old of q above which (5) is binding and below which is non-
+binding at the optimum. In the case of
+� ¯δ
+δ q∗(δ)f(δ)dδ > q
+which indicates that (5) is inactive, the optimal solution
+{p∗(δ), q∗(δ)} to (OP) will be the same as the one to (OP)
+without considering the reputation constraint. To this end, we
+have the following approach to address (OP) for given q. First,
+we solve (OP) without considering the reputation constraint.
+If the obtained solution satisfies the reputation constraint, then
+it is optimal to (OP). Otherwise, we replace the constraint (5)
+in (OP) by
+� ¯δ
+δ
+q(δ)f(δ)dδ = q,
+(12)
+as the reputation constraint holds as an equality at the op-
+timal contract design in this regime. Solving (OP) without
+incorporating the reputation constraint is a classical optimal
+contract design problem. In this work, we focus on developing
+a systematic approach to address the second case where (12)
+is considered in the constraints.
+Remark: The reputation constraint implicitly penalizes the
+SP for not serving IoT users with low valuation (the users of
+lower types). The reason is that not serving users is equivalent
+to providing zero quality service with zero cost which indeed
+decreases the average QoS. This is different from the setup
+in the classical contract design in which the SP only serves
+the consumers with positive valuations (e.g., based on the
+metric called virtual valuation δ − 1−F (δ)
+f(δ) ). Our model aims
+to serve all users including those of low types that might not
+contribute to the SP’s profit. Hence, the proposed framework
+with the reputation constraint has the capability to enhance the
+accessibility and affordability of the service to all users.
+B. Optimal Contract Solution
+Based on (1), we obtain p(δ) = Φ(δ, q(δ)) − V (δ), where
+we suppress the notations q and p in V . This is reasonable
+since the payoff of an IoT user depends on its type when the
+contract is designed. Then, by regarding V (δ) as a decision
+variable instead of p(δ), the problem can be rewritten as
+(OP′) :
+max
+{q(δ),V (δ)}
+� ¯δ
+δ
+�
+Φ(δ, q(δ)) − V (δ) − C(q(δ))
+�
+f(δ)dδ
+s.t. dV
+dδ = q(δ), dq(δ)
+dδ
+≥ 0, V (δ) = 0,
+� ¯δ
+δ
+q(δ)f(δ)dδ = q.
+(OP′) can be regarded as an optimal control problem.
+Specifically, by following the notations in control theory, we
+denote u(δ) = q(δ) by the control variable and x1(δ) = V (δ)
+by the state variable. Then, we obtain ˙x1 = u(δ) with the
+initial value x1(δ) = 0. The control input admits an increasing
+property with the type parameter δ, i.e., ˙u(δ) ≥ 0.
+The remaining difficulty in solving (OP′) lies in the reputa-
+tion constraint. To facilitate the design of the optimal control
+strategy, we introduce a new state variable x2(δ) satisfying
+˙x2(δ) = u(δ)f(δ). Therefore, the reputation constraint can be
+replaced by
+˙x2(δ) = u(δ)f(δ),
+(13)
+
+with boundary values: x2(¯δ) = q and x2(δ) = 0.
+For clarity, we present the problem (OP′) with new nota-
+tions as follows:
+(OP′′) :
+max
+{u(δ),x(δ)}
+� ¯δ
+δ
+�
+Φ(δ, u(δ)) − x1(δ) − C(u(δ))
+�
+f(δ)dδ
+s.t.
+˙x1(δ) = u(δ), x1(δ) = 0,
+˙x2(δ) = u(δ)f(δ), x2(¯δ) = q, x2(δ) = 0,
+˙u(δ) ≥ 0.
+where x = [x1, x2]T . Next, by defining λ = [λ1, λ2]T , the
+Hamiltonian of (OP′′) can be expressed as
+H(x(δ), u(δ), λ(δ), δ) =
+�
+Φ(δ, u(δ)) − x1(δ)
+− C(u(δ))
+�
+f(δ) + λ1(δ)u(δ) + λ2(δ)u(δ)f(δ),
+(14)
+where λ1 and λ2 are costate variables corresponding to (10)
+and (13), respectively.
+By using the Pontryagin maximum principle [44], we can
+obtain the optimal solution (x∗(δ), u∗(δ)) by solving the
+following Hamilton system:
+H(x∗(δ), u∗(δ), λ∗(δ), δ) ≥ H(x∗(δ), u(δ), λ∗(δ), δ), (15)
+˙x∗
+1 = ∂H(x∗(δ), u∗(δ), λ∗(δ), δ)
+∂λ1(δ)
+= u∗(δ),
+(16)
+˙x∗
+2 = ∂H(x∗(δ), u∗(δ), λ∗(δ), δ)
+∂λ2(δ)
+= u∗(δ)f(δ),
+(17)
+˙λ∗
+1 = −∂H(x∗(δ), u∗(δ), λ∗(δ), δ)
+∂x1(δ)
+= f(δ),
+(18)
+˙λ∗
+2 = −∂H(x∗(δ), u∗(δ), λ∗(δ), δ)
+∂x2(δ)
+= 0,
+(19)
+λ1(¯δ) = 0,
+(20)
+λ2(¯δ) is a constant.
+(21)
+Note that (20) and (21) are boundary conditions. Specifically,
+the initial state of x1 is fixed, and we only have freedom in
+specifying boundary condition at the terminal time. Then, the
+corresponding costate variable λ1 at the time ¯δ should equal to
+the derivative of the terminal payoff with respect to the state
+x1 at ¯δ based on the maximum principle. Since the objective
+function in (OP′′) does not include the terminal payoff, then
+we obtain λ1(¯δ) = 0. Similarly, the initial and terminal states
+of x2 are fixed, and we can specify the boundary condition
+(21) from (19) in which λ2 admits a constant value.
+Furthermore, (15) ensures the optimality of control u∗(δ).
+Thus, using the first-order condition, (15) can be simplified as
+∂H(x∗(δ), u(δ), λ∗(δ), δ)
+∂u(δ)
+=
+�∂Φ(δ, u(δ))
+∂u(δ)
+− dC(u(δ))
+du(δ)
+�
+·f(δ) + λ∗
+1(δ) + λ∗
+2(δ)f(δ) = 0.
+(22)
+In addition, (18) and (20) indicate that
+λ∗
+1(δ) = F(δ) − 1.
+(23)
+Note that the end-point of x2 is fixed, and hence λ2(¯δ) needs
+to be determined rather than simply being 0. Based on (19),
+we obtain
+λ∗
+2(δ) = β, ∀δ ∈ ∆,
+(24)
+where β is a constant to be determined.
+We have obtained the optimal solutions for λ∗
+1(δ) and λ∗
+2(δ).
+To design the optimal u∗(δ), we next focus on the optimality
+condition (22). The distribution of user’s type can be general,
+e.g., normal, exponential, or learnt from the historical data.
+We first solve (OP′′) without considering the monotonicity
+constraint ˙u(δ) ≥ 0. Then, the obtained control uco(δ) from
+(22) is a candidate optimal solution. By plugging (23) and
+(24) into (22) and using the defined functions (8) and (9), we
+obtain dC(uco(δ))
+du(δ)
+− ∂Φ(δ,u(δ))
+∂u(δ)
+= F (δ)−1
+f(δ)
++ β which leads to
+uco(δ) = 1
+a ln
+� 1
+aσ
+�F(δ) − 1
+f(δ)
++ δ + β
+��
+.
+(25)
+The second-order condition gives
+∂2H(x∗(δ),u(δ),λ∗(δ),δ)
+∂u(δ)2
+=
+−σa2eau(δ)f(δ) < 0, and hence uco(δ) is a maximizer
+of the Hamiltonian. The maximum principle is a necessary
+condition for the optimal solution of (OP′′). Then, we further
+check the sufficient condition for optimality on the maximized
+Hamiltonian. Specifically, by verifying that the Hamiltonian
+H(x(δ), u(δ), λ(δ), δ) is concave in both x and u, the so-
+lution uco(δ) is optimal to (OP′′) without considering the
+monotonicity constraint. Indeed, based on the Mangasarian
+sufficiency theorem [45], a stronger conclusion is that the
+obtained control uco(δ) is the unique optimal solution as the
+Hamiltonian is strictly concave in u. Based on the dynamics
+in (OP′′), the optimal state trajectory is also unique.
+We next verify whether uco(δ) satisfying the monotonicity
+constraint ˙u(δ) ≥ 0. In (25), the CDF F(δ) is increasing with
+δ, but the presence of f(δ) makes the monotonicity of u(δ)
+unclear. We present the following lemma which can be proved
+using optimality condition to (25).
+Lemma 2. If 2f 2(δ)+(1−F(δ))f ′(δ) > 0, then the obtained
+solution uco(δ) is optimal. In addition, a decreasing 1−F (δ)
+f(δ)
+leads to an optimal uco(δ).
+Proof. We
+need
+to
+ensure
+that
+(25)
+is
+increasing
+with
+δ.
+The
+first-order
+condition
+of
+(25)
+gives
+�
+F (δ)−1
+f(δ)
++ δ + β
+�−1 �
+f 2(δ)−(F (δ)−1)f ′(δ)
+f 2(δ)
++ 1
+�
+>
+0.
+In the first part, β is a constant determined based on
+� ¯δ
+δ uco(δ)f(δ)dδ =
+� ¯δ
+δ
+1
+a ln( 1
+aσ[ F (δ)−1
+f(δ)
++ δ + β])f(δ)dδ = q.
+The integrand should be well-defined to make the equation
+satisfied.
+The
+existence
+of
+such
+β
+is
+guaranteed
+as
+� ¯δ
+δ ln( 1
+aσ[ F (δ)−1
+f(δ)
++ δ + β])f(δ)dδ is monotonically increasing
+in β. Thus, F (δ)−1
+f(δ) +δ+β > 0 which is ensured by the choice
+of β. Then, we need to have
+f 2(δ)−(F (δ)−1)f ′(δ)
+f 2(δ)
++ 1 > 0
+which gives the result. We can also verify that if 1−F (δ)
+f(δ)
+is
+decreasing in δ, then 2f 2(δ) + (1 − F(δ))f ′(δ) > 0 holds
+which yields the result.
+■
+Remark: The distributions of IoT user’s type satisfying the
+condition in Lemma 2 are quite general, including the uni-
+form, normal and exponential ones. Note that the distributions
+without a large and sudden decrease in the probability density
+function (PDF) f(δ) generally satisfy the condition in Lemma
+2, and hence (25) gives the optimal solution.
+
+Back to (24), the constant β can be obtained by solving
+the reputation constraint (12), i.e.,
+� ¯δ
+δ
+1
+a ln( 1
+aσ[ F (δ)−1
+f(δ)
++ δ +
+β])f(δ)dδ = q. The expression u∗(δ) characterizes the pro-
+vided sensing QoS in terms of the user’s type. We next focus
+on obtaining the pricing scheme of the sensing services. To
+this end, the expression of x∗
+1(δ) becomes critical. Based on
+(16), we obtain
+˙x∗
+1(δ) = 1
+a ln
+� 1
+aσ
+�F(δ) − 1
+f(δ)
++ δ + β
+��
+.
+(26)
+Then, x∗
+1(δ) can be determined by (26) and x∗
+1(δ) = 0.
+The following Theorem 1 explicitly characterizes the optimal
+contracts in the considered scenario.
+Theorem 1. Under the condition 2f 2(δ)+(1−F(δ))f ′(δ) > 0
+in Lemma 2, the optimal contracts {q∗(δ), p∗(δ)} designed by
+the SP are as follows:
+q∗(δ) = 1
+a ln
+� 1
+aσ
+�F(δ) − 1
+f(δ)
++ δ + β
+��
+,
+p∗(δ) = Φ(δ, q∗(δ)) − φ(δ) = δq∗(δ) − φ(δ),
+(27)
+where β is determined from
+� ¯δ
+δ q∗(δ)f(δ)dδ = q, and ˙φ(δ) :=
+1
+a ln( 1
+aσ[ F (δ)−1
+f(δ)
++ δ + β]) with φ(δ) = 0.
+Structure of the optimal contracts: The q∗(δ) in (27) can be
+naturally decomposed into three parts, and each one includes
+a term F (δ)−1
+f(δ) , δ, and β, corresponding to the incentives of
+IoT users, the utility of SP, and the reputation of service
+provision, respectively. Recall that λ∗
+1(δ) = F(δ) − 1. Thus,
+the first term quantifying the impact of IC constraint on
+q∗(δ) captures the statistics of the IoT user types. The second
+term including δ arises from the maximization of objective
+function of SP which yields him the largest revenue. The third
+constant term β indicates that the sensitivity of reputation
+constraint is the same for every type of users. This finding
+is consistent with the fact that the reputation constraint takes
+the aggregated service provision over all users into account,
+i.e., the mean QoS. The service pricing function p∗(δ) is
+characterized based on q∗(δ) through relation (1) and hence
+has a similar decomposition interpretation as q∗(δ). In sum,
+the structure of optimal contracts in Theorem 1 incorporates
+a service payoff maximization term and two adjusting terms
+for user incentives.
+IV. ANALYTICAL RESULTS OF SPECIAL CASES
+In this section, we present analytical results of optimal
+contracts for two typical distributions of the user’s type.
+A.
+Uniform User Type Distribution
+When δ is uniformly distributed, its PDF and cumulative
+density function (CDF) admit the forms: f(δ) =
+1
+¯δ−δ and
+F(δ) =
+δ−δ
+¯δ−δ, δ ∈ ∆. Based on Theorem 1, the sensing
+QoS function is q∗(δ) =
+1
+a ln( 1
+aσ[2δ − ¯δ + β]). Due to the
+logarithm function, q∗(δ) is nonlinear with δ. In addition, the
+marginal sensing QoS is decreasing with the IoT user’s type.
+One reason is that increasing sensing QoS is harder in large q
+regime than its counterpart for the SP. Further, the unknown
+constant β in (24) can be solved from
+�
+β−¯δ
+2
++ ¯δ
+�
+ln(β + ¯δ)−
+�
+β−¯δ
+2
++ δ
+�
+ln(β − ¯δ + 2δ) = (aq + ln(aσ) + 1)(¯δ − δ).
+The optimal sensing pricing scheme in the contract is
+characterized in the following corollary.
+Corollary 2. Under the uniform distribution of the IoT user’s
+type δ, the price of sensing service is equal to p∗(δ) =
+δq∗(δ) +
+δ−δ
+a(¯δ−δ)(ln(aσ) + 1) −
+1
+a(¯δ−δ)
+� �
+β−¯δ
+2
++ δ
+�
+ln(β − ¯δ +
+2δ) −
+�
+β−¯δ
+2
++ δ
+�
+ln(β − ¯δ + 2δ)
+�
+.
+B. Exponential User Type Distribution
+When the user’s type δ admits the exponential distribution,
+then the number of IoT users with mission-critical tasks is less
+than the ones with nonmission-critical tasks. Specifically, the
+PDF and CDF of δ with rate ρ are equal to f(δ) = ρe−ρδ and
+F(δ) = 1−e−ρδ, respectively. Then, the optimal sensing QoS
+function has the form q∗(δ) = 1
+a ln( 1
+aσ[δ − 1
+ρ + β]), where β
+can be computed from
+� ¯δ
+δ ln( 1
+aσ[δ − 1
+ρ + β])ρe−ρδdδ = aq.
+Similar to the uniform distribution scenario, we can char-
+acterize the optimal pricing as follows.
+Corollary 3. Under the exponential distribution of the user’s
+type δ, the optimal pricing of the sensing service in the
+contract is p∗(δ) = δq∗(δ) − 1
+a(δ + β − 1
+ρ) ln(δ + β −
+1
+ρ) + δ
+a (1 + ln(aσ)) − γ, where the constant γ is equal to
+γ = δ
+a(1 + ln(aσ)) − 1
+a(δ + β − 1
+ρ) ln(δ + β − 1
+ρ).
+We elaborate more on exponential distribution scenario in
+case studies in Section VII. In other cases with more general
+distributions of the IoT user’s type, we can directly apply
+Theorem 1 to obtain the optimal SaaS contracts. However,
+note that the support of f(δ) needs to be consistent with the
+range of δ. Hence, if a normal distribution is used, it needs to
+be truncated in order to be compatible with the framework.
+V. COMPARISON TO THE BENCHMARK SCENARIO
+Under the full information scenario, the sensing SP knows
+the type of each IoT user. Thus, the IC constraint (3) becomes
+no longer necessary. Then, the optimal contract design prob-
+lem for SaaS becomes:
+(OP − B) :
+max
+{q(δ),V (δ)}
+� ¯δ
+δ
+�
+p(δ) − C(q(δ))
+�
+f(δ)dδ
+s.t. V (δ) ≥ 0, ∀δ,
+� ¯δ
+δ
+q(δ)f(δ)dδ = q.
+Next, we solve (OP − B) from an optimal control perspec-
+tive again, and the results are summarized in Theorem 2. For
+clarity, we denote by qb(δ), V b(δ) the optimal solutions to
+(OP − B). Further analysis indicates that V b(δ) = 0, ∀δ, and
+the pricing scheme is charaterized by pb(δ) = Φ(δ, qb(δ)).
+By regarding q(δ) as a control variable, i.e., q(δ) = u(δ),
+and introducing a state ˙x(δ) = u1(δ)f(δ) with boundary
+
+constraints x(¯δ) = q and x(δ) = 0, we can reformulate
+(OP − B) as
+(OP − B′) : max
+{u(δ)}
+� ¯δ
+δ
+�
+Φ(δ, u(δ)) − C(u(δ))
+�
+f(δ)dδ
+s.t. ˙x(δ) = u(δ)f(δ), x(¯δ) = q, x(δ) = 0.
+Note that (OP − B′) is an optimal control problem with
+fixed initial and terminal state constraints. The Hamiltonian
+of (OP − B′) is
+H(x(δ), u(δ), λ(δ), δ) =
+�
+Φ(δ, u(δ)) − C(u(δ))
+�
+·f(δ) + λ(δ)u(δ)f(δ),
+(28)
+where
+λ
+is
+the
+costate
+variable
+associated
+with
+the
+state
+dynamics.
+The
+maximum
+principle
+yields
+the
+following Hamilton system: H(xb(δ), ub(δ), λb(δ), δ)
+≥
+H(xb(δ), u(δ), λb(δ), δ), ˙xb = ub(δ)f(δ), ˙λb = 0, λ(¯δ) = β,
+where β is a real constant.
+The
+first-order
+condition
+of
+(28)
+with
+respect
+to
+u
+is
+∂H(xb(δ),u(δ),λb(δ),δ)
+∂u
+=
+( ∂Φ(δ,u(δ))
+∂u
+−
+dC(u(δ))
+du
+)f(δ) +
+λb(δ)f(δ) = 0. Further, the second-order conditions of Hamil-
+tonian (28) with respective to x and u are nonpositive, and
+hence the obtained ub(δ) is optimal. Then, the optimal control
+ub(δ) satisfies (δ − aσeaub(δ) + β)f(δ) = 0, which further
+yields ub(δ) = 1
+a ln δ+β
+aσ . The constant β can be solved from
+� ¯δ
+δ ub(δ)f(δ)dδ = q.
+We summarize the optimal contract for SaaS under the
+complete information in the following theorem.
+Theorem 2. When the SP has the complete incentive infor-
+mation of the IoT users, the optimal contracts {qb(δ), pb(δ)}
+are designed as follows:
+qb(δ) = 1
+a ln
+�δ + β
+aσ
+�
+,
+pb(δ) = Φ(δ, qb(δ)) = δqb(δ),
+(29)
+where β is determined from
+� ¯δ
+δ ln δ+β
+aσ f(δ)dδ = aq.
+Remark: Theorem 2 helps to identify the fundamental dif-
+ferences of optimal contracts designed under complete and in-
+complete information structures. Comparing with the designed
+optimal contracts {q∗(δ), p∗(δ)} in Theorem 1, the sensing
+QoS mapping qb(δ) and pricing function pb(δ) in Theorem 2
+do not contain terms F (δ)−1
+f(δ)
+≤ 0 and φ(δ) ≥ 0, respectively.
+The different values of β in q∗(δ) and qb(δ) prohibit the
+conclusion that q∗(δ) ≤ qb(δ) and p∗(δ) ≤ pb(δ). Note that
+the constraint
+� ¯δ
+δ q(δ)f(δ)dδ = q indicates the same mean
+QoS in two scenarios without/with asymmetric information
+between SP and users. Therefore, when q∗(δ) ̸= qb(δ),
+∀δ ∈ [δ, ¯δ], we can conclude that there exists at least one ˜δ
+where q∗(˜δ) = qb(˜δ), and the IoT users in the benchmark case
+pay more for the service due to φ(δ) ≥ 0, i.e., p∗(˜δ) < pb(˜δ).
+Another remark is that the total profit of sensing SP by
+providing the optimal contracts resulting from (OP − B) is
+no less than the one from (OP) due to the removal of IC
+constraint which enlarges the feasible decision space. The
+profit difference can be interpreted as the private user’s type
+information cost which we will quantify in Section VII.
+VI. OPTIMAL CONTRACTS FOR GENERAL USER’S TYPE
+DISTRIBUTIONS
+In this section, we investigate the scenarios when the density
+condition in Lemma 2 does not hold. We provide an alternative
+maximum principle and a full characterization of optimal
+contracts for SaaS in this general case.
+A. Maximum Principle and Optimality Analysis
+Following the notations in (OP′′) except replacing u with
+x3 and introducing a new control variable µ, we formulate the
+following problem:
+(OP − E) :
+max
+{µ(δ),x1(δ),
+x2(δ),x3(δ)}
+� ¯δ
+δ
+�
+Φ(δ, x3(δ)) − x1(δ) − C(x3(δ))
+�
+f(δ)dδ
+s.t.
+˙x1(δ) = x3(δ), x1(δ) = 0,
+˙x2(δ) = x3(δ)f(δ), x2(¯δ) = q, x2(δ) = 0,
+˙x3(δ) = µ(δ), µ(δ) ≥ 0.
+Note that (OP − E) is an optimal control problem with
+three state variables x1, x2, x3 and a control variable µ, where
+the initial points of x1 and x2, and the boundary points of x2
+are fixed.
+The
+Hamiltonian
+of
+(OP − E)
+can
+be
+written
+as
+H(x(δ), µ(δ), λ(δ), δ) = [Φ(δ, x3(δ)) − x1(δ) − C(x3(δ))] ·
+f(δ) + λ1(δ)x3(δ) + λ2(δ)x3(δ)f(δ) + λ3(δ)µ(δ), where
+x = [x1, x2, x3]T and λ = [λ1, λ2, λ3]T . To differentiate
+with the optimal solution (x∗(δ), u∗(δ)) in Theorem 1, we
+denote by (xo(δ), µo(δ)) the optimal solution to the cases
+with general user’s type distribution. Using the Pontryagin
+maximum principle, we obtain (xo(δ), µo(δ)) by solving the
+Hamilton system:
+H(xo(δ), µo(δ), λo(δ), δ) ≥ H(xo(δ), µ(δ), λo(δ), δ), (30)
+˙xo
+1 = ∂H(xo(δ), µo(δ), λo(δ), δ)
+∂λ1(δ)
+= xo
+3(δ),
+(31)
+˙xo
+2 = ∂H(xo(δ), µo(δ), λo(δ), δ)
+∂λ2(δ)
+= xo
+3(δ)f(δ),
+(32)
+˙xo
+3 = ∂H(xo(δ), µo(δ), λo(δ), δ)
+∂λ3(δ)
+= µo(δ),
+(33)
+˙λo
+1 = −∂H(xo(δ), µo(δ), λo(δ), δ)
+∂x1(δ)
+= f(δ),
+(34)
+˙λo
+2 = −∂H(xo(δ), µo(δ), λo(δ), δ)
+∂x2(δ)
+= 0,
+(35)
+˙λo
+3 = −���H(xo(δ), µo(δ), λo(δ), δ)
+∂x3(δ)
+= −
+�∂Φ(δ, x3(δ))
+∂x3(δ)
+− dC(x3(δ))
+dx3(δ)
+�
+f(δ)
+− λo
+1(δ) − λo
+2(δ)f(δ),
+(36)
+λ1(¯δ) = 0,
+(37)
+λ2(¯δ) is a constant,
+(38)
+λ3(δ) = λ3(¯δ) = 0.
+(39)
+Note that (37) and (38) are boundary conditions which are
+similar to the ones in (20) and (21). In (OP − E), we include
+
+another state variable x3 which does not have initial and
+terminal constraints. Then, based on the maximum principle
+[44], the corresponding costate variable λ3 at time δ and
+¯δ should equal to the derivative of the initial and terminal
+payoff with respect to the state x3, respectively. In (OP − E),
+the objective function does not contain individual initial and
+terminal utilities, and thus we obtain condition (39).
+First, similar to (23) and (24), we observe that
+λo
+1(δ) = F(δ) − 1,
+(40)
+λo
+2(δ) = β,
+(41)
+where the constant β can be determined using(12) after the
+QoS mapping qo(δ) is characterized.
+In addition, by integrating (36), we obtain
+λo
+3(δ) = −
+� δ
+δ
+�∂Φ(δ, x3(δ))
+∂x3(δ)
+− dC(x3(δ))
+dx3(δ)
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ.
+(42)
+Using the transversality conditions λ3(δ) = λ3(¯δ) = 0
+yields λ3(¯δ) = −
+� ¯δ
+δ ( ∂Φ(δ,x3(δ))
+∂x3(δ)
+− dC(x3(δ))
+dx3(δ) )f(δ) + λo
+1(δ) +
+λo
+2(δ)f(δ)dδ = 0. Furthermore, (30) indicates that µo(δ)
+maximizes H with µo(δ) ≥ 0. Note that in the Hamil-
+tonian H, the last term λ3(δ)µ(δ) imposes a non-positive
+value constraint on λ3(δ). Otherwise, H is unbounded from
+above due to µ(δ) ≥ 0. Then, to ensure the feasibility of
+maximization, we have λ3(δ) ≤ 0 which is equivalent to
+� δ
+δ ( ∂Φ(δ,x3(δ))
+∂x3(δ)
+−C′(x3(δ)))f(δ) +λo
+1(δ)+λo
+2(δ)f(δ)dδ ≥ 0.
+Thus, when λ3(δ) < 0, ˙xo
+3(δ) = µo(δ) = 0. Therefore, the
+complementary slackness condition can be written as follows,
+∀δ ∈ [δ, ¯δ],
+˙xo
+3(δ)
+� δ
+δ
+�∂Φ(δ, xo
+3(δ))
+∂xo
+3(δ)
+− dC(x3(δ))
+dx3(δ)
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0.
+(43)
+We can verify that the maximum principle (30)–(39) is also
+sufficient for optimality as the associated Hamiltonian equation
+is concave in both x and µ. Furthermore, the Hamiltonian is
+strictly concave in x3 and other states are uniquely determined
+by x3. Thus, the optimal control and optimal state trajectory
+are unique [45]. We next explicitly characterize this optimal
+solution.
+B. Characterization of Optimal Contracts
+We next analyze the optimal contracts in two regimes
+regarding ˙xo
+3(δ), i.e., ˙xo
+3(δ) > 0 and ˙xo
+3(δ) = 0. Based on
+(43), in the interval of δ that ˙xo
+3(δ) > 0, then λo
+3(δ) = 0
+for all δ in this interval, which further indicates ˙λo
+3 = 0.
+Hence, from (36), the following equation holds: ( ∂Φ(δ,x3(δ))
+∂x3(δ)
+−
+dC(x3(δ))
+dx3(δ) )f(δ) + λo
+1(δ) + λo
+2(δ)f(δ) = 0, which is exactly
+the same maximality condition presented in (22), where x3(δ)
+plays the role as u(δ). Following the same analysis in Section
+III-B, the optimal solutions to xo
+1, xo
+2, xo
+3, λo
+1 and λo
+2 in
+Hamilton system (30)–(39) coincide with x∗
+1, x∗
+2, u∗, λ∗
+1 and
+λ∗
+2 in Hamilton system (15)–(21). Thus, we can conclude that
+if xo
+3(δ) is strictly increasing over some interval and recall
+the notation x3(δ) = q(δ), the solution qo(δ) in this section
+should be the same as the one q∗(δ) in Theorem 1.
+In the other regime of ˙xo
+3(δ) = 0, xo
+3(δ) is unchanged.
+Then, the remaining task is to determine the intervals of δ in
+which qo(δ) admits a constant, and hence the service price is
+nondiscriminative. Note that these intervals definitely include
+the ones when q∗(δ) is decreasing, i.e., the monotonicity con-
+straint of sensing QoS is violated. For notational convenience,
+let [δ1, δ2] be the interval when qo(δ) is a constant, δ ∈ [δ1, δ2].
+We know that for δ < δ1 and δ > δ2, qo(δ) is increasing, and
+thus ˙xo
+3(δ) > 0. Based on (43), we obtain condition λo
+3(δ) = 0.
+Since the costate variable λo
+3 is continuous, then at the critical
+points δ1 and δ2, λo
+3(δ1) = λo
+3(δ2) = 0, and using (42) yields
+� δ2
+δ1
+�∂Φ(δ, q(δ))
+∂q(δ)
+− dC(q(δ))
+dq(δ)
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0.
+(44)
+To this end, we discuss three possible cases that qo(δ)
+is nondiscriminative over δ ∈ [δ1, δ2] subsequently. When
+analyzing qo(δ), we constantly refer to the optimal solution
+q∗(δ) in Theorem 1. Besides, we assume that both λo
+1 and λo
+2
+are known through (40) and (41) with an exception of β to be
+specified later.
+Case I: (δ1 = δ). In this case, (44) is reduced to
+� δ2
+δ
+�∂Φ(δ, q1)
+∂q(δ)
+− dC(q1)
+dq(δ)
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0,
+q1 = q∗(δ2).
+(45)
+One illustrative example for this scenario is shown in Fig.
+3(a), where for δ ∈ [δ2, ¯δ], qo(δ) = q∗(δ). In addition, the
+constant value q1 is no greater than q∗(δ), i.e., q1 ≤ q∗(δ).
+We prove this result by contradiction. If q1 > q∗(δ), then
+q1 > q∗(˜δ) for any ˜δ close enough to δ. Along with the entire
+trajectory q∗(δ), we introduce a virtual variable λ∗
+3(δ) which
+is a counterpart of λo
+3(δ), and thus we have λ∗
+3(δ) = 0. Recall
+the notation x3 = q, and then the partial integrand ∂Φ(δ,q(δ))
+∂q(δ)
+−
+dC(q(δ))
+dq(δ)
+in (42) decreases when the value of q increases due
+to the convexity of cost function C. Thus, the entire λo
+3(δ)
+increases if q becomes larger. Therefore, for ˜δ close enough to
+δ and based on the assumption q1 > q∗(δ), we obtain λo
+3(˜δ) >
+λ∗
+3(˜δ) = 0, contradicting the condition λo
+3(δ) ≤ 0, ∀δ ∈ [δ, ¯δ].
+Therefore, we can obtain δ2 and the corresponding value q1
+by solving two equations in (45).
+Case II: (δ < δ1 < δ2 < ¯δ). When the interval [δ1, δ2] lies
+in the interior of the entire regime δ, (44) becomes
+� δ2
+δ1
+�∂Φ(δ, q2)
+∂q(δ)
+− dC(q2)
+dq(δ)
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0,
+q2 = q∗(δ1) = q∗(δ2).
+(46)
+We can solve for two unknowns δ1 and δ2 based on (46), and
+subsequently we obtain q2. Case II is depicted in Fig. 3(b).
+
+𝛿
+𝑞∗(𝛿)
+𝑞𝑜(𝛿)
+𝛿
+𝛿=𝛿1
+𝛿2
+𝑞1
+𝑞𝑜 𝛿 , 𝑞∗(𝛿)
+𝑞𝑜(𝛿)
+Case I: 𝛿=𝛿1
+(a) Case I: δ1 = δ
+𝛿
+𝑞∗(𝛿)
+𝑞𝑜(𝛿)
+𝛿
+𝛿
+𝛿2
+𝑞2
+𝑞𝑜 𝛿 , 𝑞∗(𝛿)
+𝑞𝑜(𝛿)
+𝛿1
+𝑞𝑜 𝛿 , 𝑞∗(𝛿)
+Case II: 𝛿<𝛿1<𝛿2<𝛿
+(b) Case II: δ < δ1 < δ2 < ¯δ
+𝛿
+𝑞∗(𝛿)
+𝑞𝑜(𝛿)
+𝛿=𝛿2
+𝛿
+𝛿1
+𝑞3
+𝑞𝑜 𝛿 , 𝑞∗(𝛿)
+𝑞𝑜(𝛿)
+Case III: 𝛿=𝛿2
+(c) Case III: δ2 = ¯δ
+Fig. 3. In all three figures, qo(δ) and q∗(δ) represent the QoS of SaaS with and without considering the monotonicity constraint, respectively. In addition,
+the optimal solution qo(δ) coincides with q∗(δ) over some interval except δ ∈ [δ1, δ2] in three cases. For δ ∈ [δ1, δ2], qo(δ) is nondiscriminative and admits
+constant values q1 q2 and q3 in (a), (b) and (c), respectively.
+Case III: (δ2 = ¯δ). When δ2 coincides with the end-point
+¯δ, (44) can be written as
+� ¯δ
+δ1
+�∂Φ(δ, q3)
+∂q(δ)
+− dC(q3)
+dq(δ)
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0,
+q3 = q∗(δ1).
+(47)
+Fig. 3(c) presents an example of case III. Similar to the
+analysis in Case I, the value of q3 satisfies q3 ≥ q∗(¯δ).
+Furthermore, δ1 and q3 can be obtained by solving (47).
+Note that in the optimal contracts, the intervals over which
+qo(δ) admitting a constant value can be a combination of the
+three cases, and there could exist multiple interior intervals
+as the one shown in Fig. 3(b). Another essential point is to
+determine λo
+2 = β in (45)–(47). As the analysis in Section
+III-B, the unknown constant β can be derived using the
+constraint (12). However, (12) needs a full expression of
+optimal qo beforehand. Therefore, two procedures including
+the derivation of optimal solution qo from (45)–(47) and the
+obtaining λ2(δ) = β by (12) are intertwined. To design the
+optimal qo(δ), we thus should solve the equations (45)–(47)
+together with (12) in a holistic manner. With derived qo(δ),
+the service pricing function po(δ) then can be characterized
+with similar steps in Section III-B.
+We summarize the optimal contracts for SaaS under general
+user’s type distribution in the following theorem.
+Theorem 3. For a general user’s type distribution f(δ) where
+2f 2(δ) + (1 − F(δ))f ′(δ) > 0 does not hold, the optimal
+contracts {qo(δ), po(δ)} designed by the SP are detailed as
+follows. The QoS mapping qo(δ) is piecewise continuous and
+weakly increasing over δ ∈ [δ, ¯δ].
+1) qo(δ) and po(δ) coincide with q∗(δ) and p∗(δ) in
+Theorem 1 except on a finite number N of disjoint
+intervals In = (δn
+1 , δn
+2 ), for n = 1, ..., N, and δn
+1 and
+δn
+2 increase with n. Furthermore,, qo(δ) = qn, ∀δ ∈ In.
+2) For the interior interval In where δn
+1 ̸= δ and δn
+2 ̸= ¯δ,
+the optimal qo(δ) satisfies
+� δn
+2
+δn
+1
+�∂Φ(δ, qn)
+∂q
+− dC(qn)
+dq
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0,
+qn = q∗(δn
+1 ) = q∗(δn
+2 ).
+(48)
+3) If δ1
+1 = δ, i.e., the interval I1 starts with δ, then the
+optimal qo(δ) satisfies
+� δ1
+2
+δ
+�∂Φ(δ, q1)
+∂q
+− dC(q1)
+dq
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0,
+q1 = q∗(δ1
+2) ≤ q∗(δ).
+(49)
+4) If δN
+2
+= ¯δ, i.e., the interval IN ends with ¯δ, then the
+optimal qo(δ) satisfies
+� ¯δ
+δN
+1
+�∂Φ(δ, qN)
+∂q
+− dC(qN)
+dq
+�
+f(δ)
++λo
+1(δ) + λo
+2(δ)f(δ)dδ = 0,
+qN = q∗(δN
+1 ) ≥ q∗(¯δ).
+(50)
+5) Based on (48)–(50) and together with (12), (40), (41),
+qn, δn
+1 and δn
+2 , n = 1, ..., N, can be computed. After
+obtaining the sensing QoS function qo(δ), the optimal
+pricing po(δ) can be derived via the relation
+po(δ) = Φ(δ, qo(δ)) − φ(δ),
+(51)
+where ˙φ(δ) = qo(δ) with φ(δ) = 0.
+Remark: For the intervals where qo(δ) = q∗(δ), po(δ) is
+monotonically increasing. For δ ∈ In, n = 1..., N, qo(δ) is a
+constant and then ˙qo(δ) = 0. Based on (51) and Φ(δ, qo(δ)) =
+δqo(δ), we obtain ˙po(δ) = δ ˙qo(δ) + qo(δ) − ˙φ(δ) = 0.
+Therefore, IoT users with a type lying in the same interval
+In, n = 1, ..., N, are provided with a menu of contracts with
+the same quality of sensing data as well as the service price.
+C. Some Analytical Results
+We end up this section by presenting analytical results on
+the pricing of sensing services. These results give insights on
+
+the obtained solutions, and they also contribute to the design
+of practical market-based contracts.
+(1) Structure of the optimal contracts: Comparing with the
+optimal contracts in Theorem 1, the ones in Theorem 3 have
+an additional feature of nondiscriminative service intervals.
+Specifically, in addition to the profit maximization and service
+reputation construction of SP, the IC constraints of users are
+completely considered in the contracts, where the additional
+monotonicity part is reflected by (48)–(50). Note that the
+nondiscriminative pricing reduces the diversity of service
+provisions to the IoT users which has an interpretation that
+the SP treats heterogeneous users equally. Different with the
+contracts in Theorem 1 of full separation, the pooling behavior
+(users of different types are offered with the same contract) in
+Theorem 3 due to irregular type distribution is to ensure the
+incentive compatibility of designed optimal contracts.
+(2) Number of intervals with nondiscriminative pricing: Fig.
+3 shows that the intervals with a decreasing q∗(δ) are included
+in In, n = 1, ..., N. Then, N is equal to the number of peaks
+(local maximum) of q∗(δ). Based on Theorem 1, we analyze
+the monotonicity of F (δ)−1
+f(δ)
++ δ, indicating that the number
+of nondiscriminative pricing regimes N coincides with the
+number of intervals where 2f 2(δ) + (1 − F(δ))f ′(δ) takes
+a negative value.
+(3) Nondiscriminative pricing for all users: When q∗(δ) is
+decreasing over δ ∈ [δ, ¯δ], then based on Theorem 3, the opti-
+mal service pricing qo(δ) is nondiscriminative for all types of
+users. In this scenario, we obtain 2f 2(δ)+(1−F(δ))f ′(δ) < 0
+for all δ. From Lemma 2, an equivalent condition is that
+1−F (δ)
+f(δ)
+increases over δ. We summarize the results in the
+following lemma.
+Lemma 3. The optimal contracts {qo(δ), po(δ)} are nondis-
+criminative for all δ if 1−F (δ)
+f(δ)
+increases over δ ∈ [δ, ¯δ]. An
+alternative equivalent condition leading to the results is that
+function log[1 − F(δ)] is strictly convex.
+Some typical distributions satisfying Lemma 3 are worth
+highlighting. One example is when f(δ) is a gamma distri-
+bution for parameter α < 1, i.e., f(δ) =
+ψαδα−1 exp(−ψδ)
+Γ(α)
+,
+where δ ≥ 0 and Γ(δ) is a complete Gamma function.
+Another example is when f(δ) admits a Weibull distribution
+under α < 1, i.e., f(δ) = ψαδα−1 exp(−ψδα), δ ≥ 0.
+In both types of distributions, most of the IoT users are
+with type δ = 0 or close to δ, and its number decreases
+exponentially as the parameter δ increases. Therefore, the SP
+designs nondiscriminative contracts for all users, extracting
+the profits from the majority of customers in the market.
+Moreover, this nondiscriminative service provision mechanism
+aligns with the phenomenon of focusing on the majority, where
+the small group of users with larger types are treated in a
+homogeneous manner as the major population nested in lower
+types.
+(4) Invariant nondiscriminative service pricing: One natural
+question is the impact of convexity of log[1 − F(δ)] on the
+service price. For various type distributions f(δ) satisfying the
+condition in Lemma 3, we show that the convexity of F(δ)
+has no influence on the neutral service pricing. Specifically,
+based on the constraint
+� ¯δ
+δ qo(δ)f(δ)dδ = q, where qo(δ) =
+qc, ∀δ, we obtain qc � ¯δ
+δ f(δ)dδ = q. Therefore, under the the
+nondiscriminative pricing of sensing services, the QoS is qc =
+q for all users. Furthermore, the IR constraint V (δ) = 0 leads
+to the optimal constant pricing pc = δq. Hence, whenever the
+SP offers a nondiscriminative price scheme to all IoT users,
+the price must be invariant equaling to δq in spite of the user’s
+type distributions.
+VII. CASE STUDIES: UAV-ENABLED VIRTUAL REALITY
+In this section, we apply the SaaS paradigm to UAV-enabled
+virtual reality as depicted in Fig. 1 to illustrate the optimal
+contract design principles. We envision a large VR service
+market in the future, and thus a huge number of users will
+purchase the VR services. This SaaS paradigm can be also
+applied to other personalized data related service provision
+scenarios, such as virtual tourism. This virtual service modality
+becomes popular under the current disruptions caused by
+COVID-19 pandemic worldwide.
+A. UAV-Enabled VR Setting
+The VR quality can be quantified by user experience related
+metrics, including the resolution of the captured scene of
+UAV (˜q1), the delay in sensing data transmission (˜q2), and
+the reliability of UAV communicating with the tower (˜q3).
+Specifically, for the resolution quality ˜q1, it can be in the
+general classes of 240p, 360p, 480p, 720p, 1080p (commonly
+available options such as in the streaming services), and the
+qualities between these classes. The delay ˜q2 is composed of
+factors including processing delay, queuing delay, transmission
+delay, and propagation delay of sensing data. The delay can
+be reduced by using a dedicated network that streamlines the
+network path, which is more costly for the sensing service
+provider. The tolerable end-to-end delay of modern VR ap-
+plications is of an order of milliseconds, and a desired QoS
+has it less than 1 or 2 milliseconds [46]. The communication
+reliability ˜q3 between UAV and tower can be measured by
+the success rate that data packets are transmitted. According
+to a video QoS tutorial by Cisco [47], the reliability should
+be above 99% for a high QoS, and it is between 99.5% and
+95% depending on the specific type of services. The reliability
+above is quantified by the packet loss rate.
+We can aggregate these major metrics into a single measure
+q taking values in the real space. More specifically, the QoS
+q can be determined by a linear combination in a form of
+κ1˜q1 + κ2˜q2 + κ3˜q3, where κi, i = 1, 2, 3, are positive
+weighting factors. Equal weighting refers to the scenario with
+κ1 = κ2 = κ3 = 1/3. To differentiate the delivered services
+and pricing in terms of metrics considered, we consider that,
+comparing with a small q, a larger q has all higher values in
+˜q1, ˜q2, and ˜q3. This modeling also fits the real-world scenario
+well, as the customers choose a higher QoS should receive
+better service in every factor considered (resolution, delay,
+reliability) by paying more service fee. We anticipate a large
+VR service market in the future, and thus a huge number of
+users will purchase the VR services. We further specify the
+mean QoS q = 5. As the sensing QoS is a mapping considering
+
+various metrics, we set the mean QoS q = 5 corresponding
+to the service with 720p resolution, 0.15sec delay, and 97%
+UAV transmission reliability. After obtaining the QoS in the
+optimal contract later on, we can reversely map q to the three
+specific metrics considered. Based on the current technologies
+in communication and VR, we consider the resolution, delay,
+and reliability admit a value from 240p to 1080p, 0.5 ms to 5
+ms, and 0.95% to 0.99%, respectively. Note that in the optimal
+mechanism design, higher types of users receive better quality
+of VR service from the SP.
+As depicted in Fig. 2, the user’s type distribution admits
+f(δ) = 0.952e−0.952δ, and thus F(δ) = 1 − e−0.952δ. These
+distribution functions are aligned with the market data as
+discussed in Example 1 in Section II-A.
+B. Optimal Contracts under Hidden Information
+Based on Corollary 3, we depict the optimal contracts
+of VR services in Fig. 4 with various values of a. The
+weighting factor σ admits a value of 0.16, which gives a
+reasonable comparison between the service charging fee and
+the cost of providing the service. In the cases with parameter
+a = 0.47, 0.49, 0.51, and using the results in Section IV-B,
+we obtain β = 1.14, 1.215, 1.315, respectively. With these
+selected parameters, the obtained service pricing also matches
+with the data market. One observation is that both the VR pric-
+ing and the QoS mappings are monotonically increasing with
+the user’s type, leading to an incentive compatible contract.
+Another phenomenon is that as a increases, the VR QoS is
+decreasing for a given user’s type under the regime δ > 0.47
+as shown in Fig. 4(b). The reason is that a larger a indicates
+a higher service cost of the SP which leads to a degraded VR
+QoS. Thus, the VR pricing decreases as well for a given δ as
+illustrated in Fig. 4(a). Different with the findings in regime
+δ > 0.47, the VR QoS increases with the parameter a when
+δ < 0.47, showing that a larger cost of the SP provides a
+better VR service for the customers of type δ < 0.47 while
+the customers paying less. Note that the mean VR QoS q
+stays the same for all investigated cases. Then, to maintain a
+constant reputation that the VR SP builds in the market, the
+received QoS for customers of type δ < 0.47 should increase
+with a comparing with those of δ > 0.47. This phenomenon
+also aligns with the fact that at the early stage of VR services
+promotion (a is large), the SP focuses more on the types of
+customers with a large population in the market (small δ in
+the exponential distribution), by providing a relatively better
+VR service. Based on the VR application modeling in Section
+VII-A, Fig. 4(c) presents the specific sensing QoS in terms of
+the considered resolution, delay, and reliability metrics. Under
+the the designed optimal contracts {p∗(δ), q∗(δ)}, Fig. 5 shows
+the corresponding utility of SP. As a increases which yields
+a larger service cost, the SP’s aggregate revenue decreases
+accordingly. In addition, for some small types δ close to δ,
+U(δ) can be negative. This phenomenon indicates that the SP
+makes most of the profits from the users who demand a high
+VR QoS.
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+VR user's type 
+0
+2
+4
+6
+8
+10
+12
+14
+VR pricing scheme p*( ) ($)
+a=0.47
+a=0.49
+a=0.51
+(a) VR pricing p∗(δ)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+VR user's type 
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+VR QoS q*( )
+a=0.47
+a=0.49
+a=0.51
+(b) VR QoS q∗(δ)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+VR user's type 
+400
+600
+800
+1000
+Resolution (p)
+a=0.47
+a=0.49
+a=0.51
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+VR user's type 
+0
+0.1
+0.2
+0.3
+Delay (sec)
+a=0.47
+a=0.49
+a=0.51
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+VR user's type 
+0.94
+0.96
+0.98
+1
+Reliability (%)
+a=0.47
+a=0.49
+a=0.51
+(c) VR QoS in terms of resolution, delay, and reliability
+Fig. 4. (a) and (b) illustrate the optimal pricing scheme and the corresponding
+QoS of VR, respectively. (c) depicts the specific sensing QoS in terms of
+resolution, delay, and reliability metrics.
+C. Optimal Contracts under Full Information
+For comparison, we present the optimal contracts under
+the full information based on Theorem 2 and quantify the
+information cost associated with the user’s private types. Fig.
+6 shows the optimal pricing pb(δ) and the QoS mapping qb(δ).
+Specifically, pb(δ) is larger than the counterpart p∗(δ) under
+asymmetric information. Due to the reputation constraint,
+the VR QoS qb(δ) has a similar trajectory as q∗(δ). The
+corresponding SP’s revenue is shown in Fig. 7. Similarly, a
+larger a reduces the payoff of the VR SP. Furthermore, we
+can conclude that the SP earns more by knowing the private
+user’s type information. For example, when a = 0.47, the
+average utility of serving a user is 4.4$ which is more than 4
+times larger than the one under hidden information depicted
+in Fig. 5.
+VIII. CONCLUSION
+In this paper, we have established a Sensing-as-Service
+(SaaS) framework for QoS-based data trading in the IoT
+markets using contract theory. The proposed framework is de-
+signed for massive IoT scenarios where users are characterized
+by their service requirements and sensing data available to the
+service provider (SP) is characterized by quality. Depending
+on the probability distribution of user’s QoS needs, the profit
+
+0
+1
+2
+3
+4
+VR user's type 
+-1
+0
+1
+2
+3
+4
+5
+6
+Utility U( ) ($)
+a=0.47
+a=0.49
+a=0.51
+a=0.47
+a=0.49
+a=0.51
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Average Utility of SP ($)
+Fig. 5.
+Utility of the SP under hidden information. The SP earns profits
+from the users who demand a better VR service.
+0
+1
+2
+3
+4
+VR user's type 
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+VR QoS qb( )
+a=0.47
+a=0.49
+a=0.51
+0
+1
+2
+3
+4
+VR user's type 
+0
+5
+10
+15
+20
+25
+30
+35
+VR pricing scheme pb( ) ($)
+a=0.47
+a=0.49
+a=0.51
+Benchmark Scenario
+Fig. 6.
+Optimal contracts in the benchmark scenario. The VR service pricing
+pb(δ) is larger than the counterpart p∗(δ).
+maximizing contract solutions are proposed between the SP
+and users, which admit different structures. Specifically, under
+a wide class of user’s type distributions without a large or
+sudden decrease, the data pricing scheme and QoS mapping
+are monotonically increasing with the user types. Otherwise,
+nondiscriminative pricing phenomenon is observed which re-
+duces the diversity of service provisions to the IoT users.
+Moreover, invariant pricing phenomenon can occur when the
+user’s type distribution decreases exponentially, and thus the
+service provider targets the majority of users in the market to
+maximize the profits. We have also validated our results using
+a case study based on the application of the SaaS framework to
+UAV-enabled virtual reality, where the SP makes more profit
+by providing data services to higher type users. Future work
+can expand the SaaS contract design to cases when bounded
+rationality is considered in the user’s behavior, i.e., users have
+uncertainty on their type parameters, and subsequently design
+robust contract mechanisms. Another direction is to develop
+0
+1
+2
+3
+4
+VR user's type 
+-5
+0
+5
+10
+15
+20
+25
+30
+Utility U( ) ($)
+a=0.47
+a=0.49
+a=0.51
+a=0.47
+a=0.49
+a=0.51
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+Average Utility of SP ($)
+Benchmark Scenario
+Fig. 7.
+Utility of the SP in the benchmark scenario. The SP’s revenue under
+full information is more than 4 times larger than the corresponding one under
+asymmetric information.
+an online learning approach to designing optimal contract
+solutions when the user’s type distribution is unknown to the
+SP.
+APPENDIX A
+PROOF OF LEMMA 1
+The first-order optimality condition (FOC) on (1) with
+respect to δ′ can be expressed as ∂Φ(δ,q(δ′))
+∂q(δ′)
+dq(δ′)
+dδ′ − dp(δ′)
+dδ′
+= 0.
+The IC constraint in (3) indicates that the user of type δ
+achieves the largest payoff when claiming its true type δ. Thus,
+under δ′ = δ, the FOC becomes ∂Φ(δ,q(δ))
+∂q(δ)
+dq(δ)
+dδ
+− dp(δ)
+dδ
+= 0,
+which yields the local incentive constraint (6). Similarly,
+the second-order optimality condition (SOC) can be written
+as:
+∂2Φ(δ,q(δ′))
+∂q(δ′)2
+( dq(δ′)
+dδ′ )2 + ∂Φ(δ,q(δ′))
+∂q(δ′)
+d2q(δ′)
+dδ′2
+− d2p(δ′)
+dδ′2
+≤ 0.
+Differentiating (6) with respect to δ further gives
+d2p(δ)
+dδ2
+=
+∂Φ2(δ,q(δ))
+∂q(δ)2
+( dq(δ)
+dδ )2 + ∂Φ2(δ,q(δ))
+∂q(δ)∂δ
+dq(δ)
+dδ
++ ∂Φ(δ,q(δ))
+∂q(δ)
+d2q(δ)
+dδ2 , and
+comparing it with the SOC, we obtain ∂Φ2(δ,q(δ))
+∂q(δ)∂δ
+dq(δ)
+dδ
+≥ 0.
+Together with Assumption 1, we obtain the monotonicity
+constraint (7). The next step is to show that (6) and (7) together
+imply the IC constraint (3). Assume that the IC constraint does
+not hold for at least one type of users, e.g., δ. Then, there
+exists a ˜δ ̸= δ such that Φ(δ, q(δ))−p(δ) < Φ(δ, q(˜δ))−p(˜δ),
+and hence
+� ˜δ
+δ ( ∂Φ(δ,q(τ))
+∂q(τ)
+dq(τ)
+dτ
+− dp(τ)
+dτ )dτ > 0, where we can
+check that the derivative of Φ(δ, q(τ)) − p(τ) with respect
+to τ is exactly the integrand. Then when ˜δ > δ which gives
+τ > δ, we obtain ∂Φ(δ,q(τ))
+∂q(τ)
+< ∂Φ(τ,q(τ))
+∂q(τ)
+by Assumption 1. In
+addition, (6) indicates that
+� ˜δ
+δ ( ∂Φ(τ,q(τ))
+∂q(τ)
+dq(τ)
+dτ
+− dp(τ)
+dτ )dτ = 0.
+Replacing
+∂Φ(τ,q(τ))
+∂q(τ)
+in the integrand by
+∂Φ(δ,q(τ))
+∂q(τ)
+yields
+� ˜δ
+δ ( ∂Φ(δ,q(τ))
+∂q(τ)
+dq(τ)
+dτ
+−
+dp(τ)
+dτ )dτ
+<
+0 since
+∂Φ(δ,q(τ))
+∂q(τ)
+<
+∂Φ(τ,q(τ))
+∂q(τ)
+and dq(τ)
+dτ
+> 0, and this inequality contradicts with
+the previous integral inequality. Similar analysis follows for
+the case when ˜δ < δ, and we can conclude that (6) and (7)
+imply the IC constraint (3).
+
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+

JtE1T4oBgHgl3EQfGQNW/content/tmp_files/2301.02911v1.pdf.txt

@@ -0,0 +1,821 @@
+TOWARDS EARLY PREDICTION OF NEURODEVELOPMENTAL
+DISORDERS: COMPUTATIONAL MODEL FOR FACE TOUCH AND
+SELF-ADAPTORS IN INFANTS
+Bruno Tafur
+University of Cambridge
+United Kingdom
+[email protected]
+Marwa Mahmoud
+University of Glasgow
+United Kingdom
+[email protected]
+Staci Weiss
+University of Cambridge
+United Kingdom
+[email protected]
+ABSTRACT
+Infants’ neurological development is heavily influenced by their motor skills. Evaluating a baby’s
+movements is key to understanding possible risks of developmental disorders in their growth. Previous
+research in psychology has shown that measuring specific movements or gestures such as face touches
+in babies is essential to analyse how babies understand themselves and their context. This research
+proposes the first automatic approach that detects face touches from video recordings by tracking
+infants’ movements and gestures. The study uses a multimodal feature fusion approach mixing spatial
+and temporal features and exploits skeleton tracking information to generate more than 170 aggregated
+features of hand, face and body. This research proposes data-driven machine learning models for
+the detection and classification of face touch in infants. We used cross dataset testing to evaluate
+our proposed models. The models achieved 87.0% accuracy in detecting face touches and 71.4%
+macro-average accuracy in detecting specific face touch locations with significant improvements over
+Zero Rule and uniform random chance baselines. Moreover, we show that when we run our model to
+extract face touch frequencies of a larger dataset, we can predict the development of fine motor skills
+during the first 5 months after birth.
+Keywords Computer Vision · Autoencoders · Neurodevelopment factors
+1
+Introduction
+Figure 1: An overview of our proposed framework. Spatial, temporal and appearance features are extracted, then they
+are concatenated with a feature integration layer and a classification approach is used to detect and classify the infant’s
+face touch, which is subsequently used to predict the neurodevelopmental scores.
+Analysing body movements in early childhood gives insights into the infant’s neurological development, and it can play
+an essential role in determining if a baby is suffering from injuries in the nervous system or a hereditary disease [1].
+arXiv:2301.02911v1  [cs.CV]  7 Jan 2023
+
+777
+Video
+Features
+Fusion Model
+Output
+Raw frames
+Skeleton coordinates
+Face touch
+Geome-
+Appea-
+Wrists / Neck / Hips
+trical
+rance
+Mouth / Cheek / Ears / Eyes /
+ral
+ Features
+features
+Nose
+No face touch
+Face coordinates
+Feature Integration
+Ears / Eyes / Mouth / Nose
+Classification
+Temporal
+Velocity Wrist
+Neurodevelopmental
+ Velocity / Displacement
+scores
+- Feature interpolationTowards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+Also, specific movements such as face touches have been found to be crucial in the development of babies since their
+fetal age [2]. For example, face touches to sensitive areas of the face such as the mouth are frequent in gestational age
+as babies get prepared to feed. Different cultures have also shown differences in self-touch in babies, and age has also
+been established as a determinant factor [2].
+Various methods are utilised to track and measure movements in babies, including 3D motion capture, sensors and
+video cameras [3]. 3D motion capture and sensors are mostly used in laboratory settings instead of a more natural
+setting for the infant as it requires specific equipment and tools [4]. There have been few studies that utilised computer
+vision with video cameras. This method has the advantages of being highly flexible to different environments, giving
+high contextual information and being easier to interpret [3]. However, it requires more complex computations, depends
+on the camera quality, angle and movement, and is difficult to generalise to untrained cases.
+Various studies have explored the fidgety movements of babies by analysing pixel displacement in video frames [5, 6, 7].
+Recently, some studies have begun to examine more robust tracking algorithms for body parts based on methods such
+as OpenPose [4, 8]. Despite being an important measure for neurodevelopment, no previous research has looked at
+specific gesture detection in much detail, such as detecting specific hand movements. Most research is centred on
+general fidgety movements or general statistics descriptors. Also, these approaches have primarily focused on a general
+classification for high-risk infants or cerebral palsy [4, 9] or classification of movement types [5, 8, 10]. In the case of
+face touches, research is limited and has centred on hand-over face gestures in adults [11] or touches from the mother to
+a child [12, 13].
+This research proposes a machine learning model for automatic detection and classification of face touches in newborn
+infants and their location around key areas of the face using features extracted from raw videos. It proposes using
+feature selection and fusion models based on temporal and spatial features, using geometric and appearance features of
+the infant’s face and body. The proposed models are validated and evaluated on a couple of datasets and then applied to
+a large video dataset to extract gesture descriptors of video of one-month-old infants automatically. Using regression,
+we demonstrate the effectiveness of extracting these gestures in predicting Mullen neurodevelopmental scores [14] for
+the same infants. To the best of our knowledge, this research represents the first study to analyse specific gestures in
+infants at this level of granularity and the first to analyse self-touch. The main contributions can be described as follows:
+• Proposing a data-driven machine learning model for detection and classification of face touch in infants
+exploiting spatial and temporal features.
+• Evaluating and validating the proposed method in a cross-dataset manner using challenging naturally collected
+datasets on infants.
+• Presenting preliminary results on using our proposed computational model to detect face touch features in
+infants on a larger labelled dataset and demonstrating the ability to predict neurodevelopmental scores of the
+infants using automatic face touch dynamics.
+• Our proposed trained validated model is available on Github as an open source tool for the community. We
+believe this work will enable future research in infant behaviour modelling and provide a tool for future
+neurodevelopmental studies.
+2
+Related Work
+The most relevant studies that use computer vision in the context of neurodevelopment analysis in infants have centered
+on tracking general movement indicators; such as aggregated data from pose coordinates [4, 8] or displacement
+information from overall images [5, 6, 7]. In the analysis of touch, a couple of studies have centred on an analysis of a
+controlled environment where a mother touches her child [12, 13].
+Infants have different proportions in their limbs than adults, making them more complex for general tracking mechanisms
+to work with the same accuracy. Therefore, the study carried out by Chambers et al. adapted tracking methods based on
+computer vision on babies [4]. Their study focused on developing a tracking method for infants’ skeleton coordinates,
+and they used these statistics to compare the risk of developing neuromotor impairment in healthy and at-risk infants.
+Their study expanded on OpenPose [15] implementation for humans by tuning the model to be used on infant videos.
+Regarding more specific movement patterns related to neurological disorders, a study by Das et al. [10] focused on
+analysing the kicking patterns of at-risk infants. Their method tracked their movements using OpenPose and extracted
+additional KAZE features [16] as image descriptors. In their experiments, the authors used an SVM classifier to
+differentiate the kicking pattern types as simultaneous movement (SM), non-simultaneous movement (NSM) and no
+movement (NM).
+2
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+In the case of touch, Chen et al. performed a couple of studies focused on the detection of interactions between a
+caregiver and a child in a controlled environment [12, 13]. Specifically, their research focused on detecting touch from
+the caregiver to the child in particular locations: head, arms, legs, hand, torso and feet. Their latest study applied two
+main methods; firstly, they extracted tracking information by detecting the skeleton locations. Secondly, they extracted
+the infant’s location in the image by applying image segmentation using the GrabCut algorithm.
+Therefore, although some studies have attempted to tackle some of these issues in infants, most of them have focused
+on the analysis of general movements instead of specific gestures. This study analyses hand to face gestures in infants
+in larger detail and granularity and proposes novel machine learning models for automatic detection.
+The relationship between face and body touch in infants and how they correlate with cognitive development has not
+been studied quantitatively and systematically before in previous literature. The Mullen Scales of Early Learning is
+used to measure the cognitive development of infants in five different categories: gross motor (GM), fine motor (FM),
+visual reception (VR), receptive language (RL) and expressive language (EL) [14, 17]. They are a key measure of the
+development of the child during the first years after birth. Previous studies have not tackled the relationship of detected
+features with MSEL scores. We aim to analyse this relationship based on gesture and movement data extracted from the
+infant.
+3
+Datasets
+For our data-driven models, we used two main datasets: BRIGHT [18] and Chambers [4]. A subset of the two datasets
+was labelled and validated by a psychology expert to be later used for our models. Then, the videos from the BRIGHT
+dataset were used to evaluate the correlations between face touch dynamics and neurodevelopmental scores.
+3.1
+BRIGHT dataset
+This dataset was provided by the ’removed for anonymous submission’ and is part of the studies carried out in the
+Brain Imaging for Global Health (BRIGHT) Project [18] in which they study infants from Gambia and UK during their
+first 24 months of life. The initial sample provided included 29 videos of UK infants. From the 29 videos, 23 videos
+were selected as some of the babies were occluded during most of the video runtime. Each video shows the behaviour
+of one infant of fewer than 2 months of age, actively responding to the input given by their mother. The videos were
+recorded in different rooms, with the infant lying down with a mirror positioned on the wall behind the head of the baby.
+The camera is static, and the infants generally cover a small portion of the frame but can be located in different parts
+of the frame. Another complex factor that characterises this dataset includes the mother’s presence during the video,
+sometimes occupying a significant part of the frame with a bigger skeleton and limbs. Also, the fact that the infants are
+lying down while the camera is facing the front means that the camera generally captures the babies’ faces from a side
+or the bottom, making them difficult to detect for traditional algorithms. The babies are shown rotated in the frames at
+different angles between 90° and -90°.
+3.2
+Chambers dataset
+This is an open dataset compiled and generated by Chambers et al [4]. 25 videos were selected based on the age of the
+infants in the video by filtering and selecting only the videos with babies less than 2 months to ensure better consistency
+with the BRIGHT dataset. The videos show babies lying down on their own and interacting in a natural environment.
+They could be dancing, playing or rolling over in their crib. The camera is sometimes moving while filming the baby,
+and the babies generally cover most of the frame. The videos do not feature other people in the frame, but the babies
+sometimes can move at different angles. Also, the resolutions are very varied between videos, with some of them being
+more blurry and with smaller frames. The babies are shown rotated in the frames at different angles between 90° and
+-90°.
+4
+Labelling
+The labelling process was carried out using a tagging system developed for this research which allowed efficient tagging
+of the image frames. Also, the tagging was carried out with the support of a psychology expert, who helped by labelling
+part of the dataset and providing her judgement about the different labelling categories.
+As this study aims to detect hand over face gestures in infants automatically, the main labelling category to tag needed
+to differentiate between face touch or no touch in each frame. Therefore, it was defined as follows:
+3
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+Table 1: Database sizes and labels
+Dataset
+Sizes
+# Videos
+25
+Total Frames
+1769
+Chambers
+Mean Frames per video
+70.76
+% On Head
+29.2%
+% Outside Head
+70.8%
+# Videos
+23
+Total Frames
+2039
+BRIGHT
+Mean Frames per video
+88.6
+% On Head
+29.7%
+% Outside Head
+70.3%
+Total Number of Frames
+3808
+• On Head: From a human perspective, it can be seen that the hand could be touching the head area. In this
+study, the head area considers any of the following locations or any area enclosed by the those locations: eyes,
+ears, nose, mouth, cheeks, forehead and neck.
+• Outside Head: From a human perspective, it can be seen that the hand is not on the head area as defined.
+Additionally, we labelled our dataset with the following non-exclusive categories: eyes, ears, nose, mouth and cheeks,
+as they are the main differentiable parts of the face. The categories were also discussed and agreed upon with the
+psychology expert to validate their significance and usefulness from the neurodevelopment perspective.
+The final labelled datasets sizes and distributions can be seen in Table I. The final proportion of “on head” versus
+“outside head” was of 29.5% to 70.5%, which is expected for this kind of natural dataset.
+5
+METHOD
+Because of the small size of the labelled dataset, we could not use an end-to-end deep learning model. In this section
+we present the feature extraction and selection steps and the proposed feature fusion machine learning model.
+5.1
+Feature extraction of face and body
+Our proposed models required spatial and temporal features related to the infants’ face touch gestures. The features
+extracted were selected considering the relationship between the hands of the baby and the face.
+5.1.1
+Extraction of face and body landmarks
+We first extracted basic face and body landmarks.
+- Pose coordinates: Positions of the skeleton parts were extracted for every baby and every frame by using the fine-tuned
+OpenPose [15] model trained by Chambers et al. [4]. Following the implementation of Chambers et al., the raw pose
+locations were normalised, smoothed and interpolated per video.
+- Face Region: Based on the extracted pose features and estimated orientation, an accurate estimate of the baby’s face
+location was carried out and the image was cropped in the face region. If no possible face was found in a given frame,
+the locations of the face of the nearest frames were used as guidance. Where possible, the face region was further
+aligned based on the locations of the eyes and nose.
+- Face coordinates: OpenPose provides general locations of the eyes, nose and ears, but its purpose is centred on getting
+the whole skeleton and not on specific facial landmarks. Therefore, information about the location of facial features
+based on 3D-FAN [19] was also used. The faces were extracted from the aligned cropped face regions.
+5.1.2
+Extraction of geometric, appearance and temporal features
+After basic landmarks features were extracted, we extracted a set of geometric and temporal feature descriptors.
+Based on the initial features, the following features were calculated:
+4
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+Figure 2: Example body skeleton extracted using OpenPose and face keypoints extracted using 3D-FAN
+- Face and body geometrical features (Distance and Angular): Based on the coordinates of the skeleton of the baby, the
+normalised distances between the wrists and the ears, eyes, neck and nose were extracted. For each case, the distances
+considered included differences in the X direction, differences in the Y direction and euclidean distances. Additionally,
+based on the coordinates of the skeleton of the baby, the angles of the elbows and shoulders were extracted.
+- Hands geometrical features (Distance): As the adapted OpenPose model by Chambers et al. [4] only generated the
+skeleton up to the wrists, additional information was obtained by extending the skeleton to the hands. The MediaPipe
+detection algorithm [20] was used in the area surrounding the wrists to obtain the hand coordinates. Based on the
+coordinates, the normalised distances between the fingers and the eyes and nose were calculated. The distances included
+differences in the X direction, differences in the Y direction and euclidean distances. Also, confidence scores were
+considered as additional features based on the confidence of the MediaPipe algorithm detecting each hand.
+- Temporal features: The temporal features were centred on aggregated information over various frames. We calculated
+features including displacement, speed and acceleration obtained based on the coordinates of the skeleton of the baby
+for the wrists and elbows.
+- Appearance Features: Histogram of Oriented Gradients (HOG) [21] is a method for feature extraction based on
+the directionality of the gradients in different locations in an image. This method has shown significant success
+rate in different image detection tasks including detecting faces and expressions [22, 23, 24] and detecting gestures
+[25, 26, 27]. These features were extracted only for the main region of interest, which is the face area. Consequently,
+these features were extracted from the cropped images of the face. Additionally, we wanted to extract more localised
+spatial information inside the face. Therefore, more granular HOG features were extracted in two specific face areas:
+one related to the upper region of the face based on the eyes location and another related to the lower region based on
+the mouth location. Also, confidence scores were considered as additional features based on the average confidence of
+the landmarks in each region as calculated by 3D-FAN.
+Figure 3: Examples of HOG features obtained for the face region, the upper head and the lower head. Note the
+challenging nature of the dataset with extreme head poses and viewpoints.
+5.1.3
+Features smoothing and data augmentation
+As a final step in the feature extraction process, we smoothed and augmented the calculated features to be able to train
+the classification models on the data. The outliers of the geometrical and temporal features per video were replaced by
+blank values, and the data was interpolated per video to cover any deleted or missing values. If data was still missing, it
+was replaced by mean values from the training data during the training stage.
+Finally, to compensate for the small size of the dataset, the training data was augmented by flipping the images
+horizontally, flipping all the features accordingly and considering the directionality of these features.
+5.2
+Face touch detection and classification
+After feature extraction and smoothing, we handled face touch detection and classification as two different classification
+problems. The first is to detect when the hand touches the face as a binary classification problem; then, we classify
+different touch location areas as a multi-label classification problem. The architectures proposed for both problems are
+5
+
+100
+200
+300
+400
+100
+200
+300
+400Input image
+Input image
+Input image
+Histogram of Oriented Gradients
+10 
+20
+20
+40
+60
+20
+100
+120
+20
+80
+100
+120
+40
+80
+60
+Histogram of Oriented Gradients
+Histogram of Oriented Gradients
+00
+20
+60
+20
+80
+100
+120
+20
+40
+60
+80
+100
+120
+20
+40
+60
+80
+100
+120
+20
+40
+60
+80
+100
+120Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+very similar. The main difference lies in the method used for the classification component in the final layer. The models
+can be divided into the following:
+5.2.1
+Feature selection and dimensionality reduction
+Our first proposed method used feature selection and dimensionality reduction and a Support Vector Machine (SVM)
+classifier for solving the face touch detection problem.
+There were four main categories of geometrical and temporal features: body distance features, hand distance features,
+angular features and temporal features. Many of the features in the same categories correlated with each other as they
+measured similar characteristics. Therefore, to ensure proper representation of the features, this method proposed
+reducing these features before training a classifier.
+Firstly, the features were filtered based on an automatic feature selection process. Random Forest was used to select
+the most representative features and prevent skewing the classifier with features that were not that significant. The
+feature selection was carried out by cross-validating with 5-folds in the training set to ensure independence. Then,
+Principal Component Analysis (PCA) was used for dimensionality reduction. PCA has shown effective results in
+detection problems when facing a large number of features [28, 29, 30]. PCA was used to filter a percentage of the
+explained variance. The threshold of this explained variance was established as a hyperparameter that was also learned
+by cross-validating with 5-folds in the training set.
+After applying PCA, the classification algorithm used SVM using an RBF kernel. The model was cross-validated
+with 5-folds in the training set to choose the best hyperparameters for SVM. The search for the best hyperparameters
+for SVM was done in combination with the search for the threshold for PCA, as the hyperparameters were possibly
+dependent on each other using a grid search method [31].
+In the case of multi-class classification for the face areas, the Label Powerset model was used with underlying SVMs
+to predict the multiple overlapping labels. Label Powerset transforms the labels by creating a class for each possible
+combination of labels and creates a classifier for each combination [32]. Consequently, it has the advantage of
+considering the possible relationships between the labels. This model was configured by tuning the hyperparameters in
+the same way as in the SVM binary classifier.
+5.2.2
+Feature optimisation using deep learned features
+Our second proposed method used autoencoders as a feature optimisation and dimensionality reduction technique.
+This method has been used in various studies as an effective way of reducing dimensionality while maintaining the
+representation of the data, and it has been successfully used before with repetitive and correlated features [33]. In this
+case, autoencoders were used to generate a latent representation of the input features.
+Firstly, the dimensions of the features were reduced based on the autoencoder model. The model uses a neural network
+architecture that learns how to represent the data in lower dimensions and reconstruct it [33]. It then minimises the error
+between the reconstruction and the original input. The aim of the autoencoder is to exploit the correlations in the input
+features to reduce the final dimensions without losing relevant information.
+This method was used with two alternatives of input features. The first one used only geometrical and temporal features.
+The second alternative also used the HOG features. The main hyperparameters that were learned for this model included
+the latent dimensions and the number of epochs. These hyperparameters were selected based on the results of a 5-fold
+cross-validation in the training set.
+After encoding the data, the classification process was done using SVM with an RBF kernel. The input features for the
+SVM classification process were the output of encoding the features with the trained encoder. The classification layer
+was also cross-validated with 5-folds in the training set to choose the best hyperparameters for SVM. Finally, in the
+case of the multi-label classification problem, the Label Powerset model was used with underlying SVMs to be able to
+predict the multiple face touch locations.
+6
+EVALUATION
+We evaluated the accuracy of the detection of face touches by using a mixture of spatial and temporal features
+and analysed models based on dimensionality reduction and optimisation techniques. The models were evaluated
+cross-dataset to validate their effectiveness and generalisation. The approaches were evaluated with three different
+configurations of the datasets to ensure the consistency of the models. Also, all segmentations of the data were grouped
+by video to ensure having different videos in each set. The three configurations used were the following:
+6
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+• Train and Cross Validate on BRIGHT dataset - Test on Chambers dataset
+• Train and Cross Validate on Chambers dataset - Test on BRIGHT dataset
+• Train and Cross Validate on Chambers and on 50% of BRIGHT dataset - Test on the other 50% of BRIGHT
+dataset
+As there was no existing baseline for these models, the models were evaluated against Zero Rule (ZeroR) baseline and
+random uniform chance. In the case of the ZeroR baseline, it is calculated by assigning the value of the majority class
+to every data point [34, 35, 36] while random chance assigns a class based on random uniform probabilities. Statistical
+McNemar’s tests were carried out to ensure the results were significantly different. The McNemar test was used as the
+compared distributions were binary targets instead of continuous variables. All the best performing models were found
+significantly different with p < 0.01 in comparison to Random Chance and Zero Rule.
+6.1
+Detection of face touches
+The main target was to determine if there was a face touch. This problem was treated as a binary classification task
+based on the classes: “on head” and “outside head”.
+The models that were analysed were the following:
+• Feature selection and dimensionality reduction based on geometrical and temporal features (RF-PCA-SVM):
+This model followed the components described in Section 5.2.1. It performed feature selection with Random
+Forest (RF) and dimensionality reduction with PCA. Finally, it performed the classification of the labels using
+SVM in the case of this binary problem. It used the geometrical distance and angular features and aggregated
+temporal features.
+• Feature optimisation using deep learned features based on geometrical and temporal features (AUTO ENC-
+SVM-I): This model was structured as described in Section 5.2.2. It used an autoencoder neural architecture to
+reduce the dimensions of the input features. Finally, it performed the classification of the labels using SVM. It
+used the geometrical features (distance and angular) and the temporal features.
+• Feature optimisation using deep learned features based on geometrical, temporal and HOG features
+(AUTOENC-SVM-II): The model was structured as described in Section 5.2.2. Similar to the previous
+model, it used an autoencoder neural architecture to reduce the dimensions of the input features and SVM for
+the binary classification problem. It used the geometrical features, temporal features and HOG features.
+The results for predicting between “on head” and “outside head” can be seen in Table 2, 3 and 4. All the models had
+significantly higher accuracy than uniform random chance and ZeroR baselines. The best performing model reached
+87% accuracy when trained in a mixture of both datasets.
+Overall the results of the three models were promising with high accuracy in comparison to the baselines. Also,
+the results were relatively similar between the three models. Some performed better on different datasets, but the
+performance was very competitive between them. All three models obtained better results than ZeroR or Random
+Chance in accuracy, precision and recall. Therefore, the results demonstrated that these models can perform well in the
+detection of face touch.
+Even though the autoencoder models (AUTOENC-SVM) outperformed the random forest and PCA model (RF-PCA-
+SVM) in two of the three dataset configurations, the difference in accuracy performance was limited. These results
+demonstrate that the RF-PCA-SVM configuration was also very effective. Possibly in larger datasets, the autoencoder
+based models could extract more representative features that could better outperform the RF-PCA-SVM model.
+Similarly, the inclusion of the HOG features in the AUTOENC-SVM-II model did not show a noticeable increase in
+performance. In the case of the BRIGHT dataset, it did show an improvement over the other models and a higher
+improvement over AUTOENC-SVM-I. However, the improvement could have been greater. This could be caused
+by the limited amount of data with very varied head poses and rotations. Therefore, the AUTOENC-SVM-II model
+might perform better if trained in larger datasets where the HOG features can be learned with more generalisable
+representations.
+Finally, even though there were various challenges in the datasets that could have a negative impact on the models’
+ability to generalise between datasets, the results demonstrated that the proposed methods had high performance in the
+detection of face touches.
+7
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+Table 2: Results -binary classification “on head” vs “outside head”.
+Training and CV dataset: Chambers. Testing dataset: Bright.
+Model
+Accuracy
+Precision
+Recall
+Test
+On Head
+On Head
+Random Chance
+50%
+29.7%
+50%
+Zero Rule
+70.3%
+0%
+0%
+RF-PCA-SVM
+80.3%
+68.7%
+62.2%
+AUTOENC-SVM-I
+80.7%
+70.8%
+59.6%
+AUTOENC-SVM-II
+80.6%
+74.4%
+53.1%
+Table 3: Results -binary classification “on head” vs “outside head”.
+Training and CV dataset: Bright. Testing dataset: Chambers.
+Model
+Accuracy
+Precision
+Recall
+Test
+On Head
+On Head
+Random Chance
+50%
+29.2%
+50%
+Zero Rule
+70.8%
+0%
+0%
+RF-PCA-SVM
+77.8%
+58.8%
+80.2%
+AUTOENC-SVM-I
+75.2%
+57.9%
+54.8%
+AUTOENC-SVM-II
+79.6%
+65.4%
+63.8%
+6.2
+Classification of face touch descriptors
+These experiments evaluate the face touch on specific locations of the face. These locations were evaluated based on the
+universe of images where there is a face touch. The key locations to predict included the following: ears, nose, cheeks,
+mouth, and eyes. The problem mas evaluated as a multi-label problem because the different classes could overlap and
+the infant could touch more than one location at the same time.
+The proposed models for this problem are the same as the ones described in Section 6.1, so we will use the same naming
+abbreviations. The main difference was the change in the classification method from SVM to Label Powerset with SVM
+[32] to tackle the problem as a multi-label classification problem. Therefore, the models that were analysed were the
+following:
+• Feature selection and dimensionality reduction based on geometrical and temporal features (RF-PCA-SVM)
+• Feature optimisation using deep learned features based on geometrical and temporal features (AUTOENC-
+SVM-I)
+• Feature optimisation using deep learned features based on geometrical, temporal and HOG features
+(AUTOENC-SVM-II)
+The experiments were carried out only on the portion of images labelled as “on head” so that it could be sufficiently
+balanced; therefore the dataset was even more limited in size than the original.
+The obtained results can be seen in Table 5, 6 and 7. The results show the macro-average accuracy, precision and recall
+of the multiple key locations per model. The highest performing model reached 71.4% average accuracy when testing
+on the Chamber’s dataset.
+Table 4: Results -binary classification “on head” vs “outside head”.
+Training and CV dataset: Chambers + 50% Bright. Testing dataset: 50% Bright.
+Model
+Accuracy
+Precision
+Recall
+Test
+On Head
+On Head
+Random Chance
+50%
+28.3%
+50%
+Zero Rule
+71.7%
+0%
+0%
+RF-PCA-SVM
+87.0%
+77.2%
+76.8%
+AUTOENC-SVM-I
+86.9%
+76.9%
+77.1%
+AUTOENC-SVM-II
+85.7%
+71.6%
+82.2%
+8
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+Table 5: Results of predicting key areasTraining and CV dataset: Chambers. Testing dataset: Bright.
+Model
+Accuracy
+Precision
+Recall
+Test
+Key Area
+Key Area
+Random Chance
+50.0%
+31.1%
+50.0%
+Zero Rule
+24.5%
+13%
+0%
+RF-PCA-SVM
+66.6%
+33.5%
+43.8%
+AUTOENC-SVM-I
+63.2%
+49.6%
+16.7%
+AUTOENC-SVM-II
+63.0%
+60.9%
+13.2%
+Table 6: Results of predicting key areasTraining and CV dataset: Bright. Testing dataset: Chambers.
+Model
+Accuracy
+Precision
+Recall
+Test
+Key Area
+Key Area
+Random Chance
+50.0%
+20.8%
+50%
+Zero Rule
+37.8%
+15%
+0%
+RF-PCA-SVM
+62.5%
+36.3%
+18.8%
+AUTOENC-SVM-I
+63.8%
+29.7%
+34.3%
+AUTOENC-SVM-II
+71.4%
+35.7%
+24.9%
+The main metric established to select the best models during cross-validation was the average macro-accuracy of the
+locations; so this was used as the main indicator of the models performance. The results demonstrated that the models
+performed effectively better than the baselines.
+As expected, the accuracies were lower than for the face touch problem as this was a complex multi-label problem
+where multiple locations can overlap on the face, and it can be difficult even for a human to determine the exact location.
+Likewise to the previous task, the results were similar between models, but these results show some indication that
+HOG features might be useful in some instances. The AUTOENCODER-SVM-II model outperformed the accuracies
+of the other models in two cases and demonstrated a considerable difference in accuracy when it was trained in the
+BRIGHT dataset. Possibly training with HOG features in more extensive and more varied datasets could make their
+representations more stable and significant in the end results.
+6.3
+Predicting neurodevelopment scores
+The next step was to evaluate our proposed model results - detected face touch dynamics of infants less than 2 months
+old - on predicting their neurodevelopmental rates collected at ages 3 and 5 months. We chose the best-performing
+model for the binary classification task (RF-PCA-SVM) and ran it on a larger dataset. Since we had the Mullen scores
+only for the BRIGHT dataset, we ran our model on an average of 490 frames per video (19 videos), a total of 9298
+frames. We then extracted the face touch frequency for each infant and evaluated it versus the Mullen Scales of Early
+Learning (MSEL) related to gross motor (GM) skills and fine motor (FM) skills. In this case, the data was limited
+because the provided metrics were evaluated per infant, and only 19 infants of the BRIGHT dataset had their information
+available.
+In the case of the MSEL metrics, the data consisted of raw scores per visit of the infant related to the different MSEL
+categories. A rate of development was calculated per infant per category based on the rate of increase during their first
+five months. The data used for this case were the gross motor (GM) skills and the fine motor (FM) skills, as they are
+related to the infant’s motor development and could be related to face touch behaviour. After calculating the rate of
+Table 7: Results of predicting key areas
+Training and cross-validation dataset: Chambers + 50% Bright. Testing dataset: 50% Bright.
+Model
+Accuracy
+Precision
+Recall
+Test
+Key Area
+Key Area
+Random Chance
+50.0%
+29.1%
+50.0%
+Zero Rule
+21.8%
+17.0%
+0%
+RF-PCA-SVM
+60.3%
+56.0%
+34.3%
+AUTOENC-SVM-I
+58.1%
+48.2%
+19.2%
+AUTOENC-SVM-II
+60.7%
+44.3%
+16.7%
+9
+
+Towards early prediction of neurodevelopmental disorders: Computational model for Face Touch and Self-adaptors in
+Infants
+development of the GM and FM skills, a correlation was calculated between the ratio of face touches per frame and
+these rates of development of each child.
+The results showed a low to moderate positive correlation between the ratio of face touches and the rate of development
+during the first months. The correlation coefficient obtained for FM was 0.599 with a significant p-value of 0.0067. The
+correlation coefficient for GM was 0.186, but the p-value was not found to be significant. It is possible that face touches
+are more related to fine motor skills as they are more specific and localised movements.
+The results indicate that measuring infants’ face touch frequencies and dynamics in their first month or two can
+be a predictive measure of their neurodevelopmental scores. It also demonstrates the effectiveness of our proposed
+computational model as a tool for the early prediction of neurodevelopmental factors. We make the trained model
+available to the research community at ’removed for anonymous submission’ as a baseline for infant’s face touch
+detection and to facilitate future research in this area on more extensive datasets.
+A dataset of 19 infants is limited, so it was not possible to test more complex prediction algorithms. However, these
+results show that, in more extensive datasets, the face touch frequency could be used as one independent variable to help
+predict infant neurodevelopment scores such as MSEL. Also, the models proposed during this research could support
+the automation of the extraction of these face touches.
+7
+Conclusion
+Our research proposed a machine learning model for automatic detection of face touches in infants using features
+extracted from videos. This is the first study to provide a computational model for detection and classification of these
+types of gestures in infants. Our proposed models using a mix of spatial and temporal features with deep learning
+features demonstrated significantly high accuracies in predicting face touch and their locations around keypoints in the
+face establishing a promising step for future research in this area. We also showed the effectiveness of the proposed
+model in predicting MSEL scores related to fine motor (FM) skills , demonstrating that our proposed model can be used
+as en early prediction tool for neurodevelopmental disorders in infants and it is considered a baseline for future work in
+this domain. We believe this research will open the door for future research in this area both on the technical as well as
+neurodevelopmetanl psychology fronts.
+Despite the promising results, there are several limitations to our model. The datasets used were recorded in almost
+uncontrolled environments, with varied camera angles and the mum’s presence in most videos. These characteristics
+made the labelling as well as the classification tasks very challenging. We are also aware of the small size of the datasets
+used in this work. Obtaining datasets, especially for infants, is a challenging task due to the privacy and ethical factors
+that need to be considered. However, we believe this research serves as a baseline for infant face touch detection and
+classification and will open the door for further research on more extensive datasets in this area.
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+12
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+arXiv:2301.13655v1  [math.CA]  31 Jan 2023
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR
+ESTIMATES
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+ABSTRACT. We develop both bilinear theory and commutator estimates in the context of
+entangled dilations, specifically Zygmund dilations (x1, x2, x3) �→ (δ1x1, δ2x2, δ1δ2x3) in
+R3. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund
+dilations and apply them to prove a paraproduct free T 1 theorem for bilinear singular in-
+tegrals invariant under Zygmund dilations. Independently, we prove linear commutator
+estimates even when the underlying singular integrals do not satisfy weighted estimates
+with Zygmund weights. This requires new paraproduct estimates.
+1. INTRODUCTION
+“Entangled” systems of dilations, see Nagel-Wainger [22], in the m-parameter product
+space Rd = �m
+i=1 Rdi have the general form
+(x1, . . . , xm) �→ (δλ11
+1
+· · · δλ1k
+k
+x1, . . . , δλm1
+1
+· · · δλmk
+k
+xm),
+δ1, . . . , δk > 0,
+and appear naturally throughout analysis. For instance, in R3 the Zygmund dilations
+(x1, x2, x3) �→ (δ1x1, δ2x2, δ1δ2x3) are compatible with the group law of the Heisenberg
+group, see e.g. Müller–Ricci–Stein [21]. Even these simplest entangled dilations are not
+completely understood, especially when it comes to the associated Calderón–Zygmund
+type singular integral operators (SIOs).
+Until recently, multiresolution methods were still missing in the Zygmund dilations
+setting, as pointed out in [5]. This was a big restriction on how to go about developing
+singular integral theory. However, the last two authors together with T. Hytönen and
+E. Vuorinen recently developed this missing Zygmund multiresolution analysis in [14].
+Such dyadic representation theorems and related multiresolution techniques had been
+highly influential in recent advances on SIOs and their applications (see e.g. [12, 13, 20,
+23]), but developing them in the entangled situation required new ideas. These tools
+then yielded very delicate weighted norm inequalities Lp(w) → Lp(w) for general non-
+convolution form Zygmund singular integrals in the optimal generality of Zygmund
+weights (introduced by Fefferman–Pipher [6])
+[w]Ap,Z := sup
+I∈RZ
+� 1
+|I|
+ˆ
+I
+w(x) dx
+�� 1
+|I|
+ˆ
+I
+w−1/(p−1)(x) dx
+�p−1
+< ∞,
+1 < p < ∞,
+2010 Mathematics Subject Classification. 42B20.
+Key words and phrases. singular integrals, multi-parameter analysis, Zygmund dilations, multiresolution
+analysis, weighted estimates.
+E.A. was supported by Academy of Finland through Grant No. 321896 “Incidences on Fractals” (PI =
+Orponen) and No. 314829 “Frontiers of singular integrals” (PI = Hytönen).
+K. L. was supported by the National Natural Science Foundation of China through project number
+12222114 and 12001400.
+1
+
+2
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+where the supremum is over Zygmund rectangles I = I1 × I2 × I3, ℓ(I3) = ℓ(I1)ℓ(I2).
+In fact, there is a precise threshold: if the kernel decay in terms of the deviation of
+z ∈ R3 from the “Zygmund manifold” |z1z2| = |z3| is not fast enough, singular integrals
+invariant under Zygmund dilations fail to be bounded with Zygmund weights. We con-
+structed counterexamples and showed the delicate positive result in the optimal range
+using the new multiresolution analysis. Previous results include [5,6,11,24].
+This rather striking threshold for weighted estimates means that it is, in particular,
+unclear in what generality natural estimates for commutators [b, T] = bT − T(b · ) hold.
+Of course, ever since the classical one-parameter result of Coifman–Rochberg–Weiss [2],
+stating that ∥[b, T]∥Lp→Lp ∼ ∥b∥BMO, commutator estimates have been a large and fun-
+damental part of the theory of SIOs and their applications. Commutator estimates in the
+Zygmund dilation setting were previously considered in [5] using the so-called Cauchy
+integral trick. That method requires weighted bounds with Zygmund weights – this is
+because it uses the fact that natural Zygmund adapted BMO functions generate Zyg-
+mund weights. But we now know [14] that such weighted bounds are quite delicate –
+and it turns out that the commutator bounds are true even in the regime where weighted
+estimates fail. We prove the following.
+1.1. Theorem. Let b ∈ L1
+loc and T be a linear paraproduct free Calderón-Zygmund operator
+adapted to Zygmund dilations as in [14]. Let θ ∈ (0, 1] be the kernel exponent measuring the
+decay in terms of the Zygmund ratio
+Dθ(x) :=
+�
+|x1x2|
+|x3|
++
+|x3|
+|x1x2|
+�−θ
+.
+Then for all such θ we have
+∥[b, T]∥Lp→Lp ≲ ∥b∥bmoZ,
+1 < p < ∞.
+As weighted estimates only hold with θ = 1, this requires a proof based on the mul-
+tiresolution decomposition [14] and a new family of “Zygmund paraproducts”. Study-
+ing paraproducts is also interesting from the technical viewpoint that, generally, proofs
+of T1 theorems display a structural decomposition of SIOs into their cancellative parts
+and paraproducts. The new Zygmund theory in [14] is designed for the fully cancellative
+case leaving out paraproducts and BMO considerations, so this is the first paper, as far as
+we know, where paraproducts are considered in the Zygmund situation. They are tricky
+objects in the entangled situation. However, while this is also a step forward towards a
+full T1 theorem in the Zygmund setting, the commutator theory that we develop does
+not require so-called partial paraproducts, and so the paraproduct tools developed here
+are not yet sufficient to prove a T1 theorem in the non-cancellative case. We also men-
+tion that during our proof we include some results of independent interest, mainly, a
+new, extremely short proof of the A∞ extrapolation theorem [3].
+Moving to a different direction, we push the Zygmund multiresolution methods [14]
+to the multilinear setting and study bilinear SIOs invariant under Zygmund dilations. A
+classical model of an n-linear SIO T in Rd is obtained by setting
+T(f1, . . . , fn)(x) = U(f1 ⊗ · · · ⊗ fn)(x, . . . , x),
+x ∈ Rd, fi : Rd → C,
+where U is a linear SIO in Rnd. See e.g. Grafakos–Torres [9] for the basic theory. Estimates
+for classical multilinear SIOs play a fundamental role in pure and applied analysis – for
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+3
+example, Lp estimates for the homogeneous fractional derivative Dαf = F−1(|ξ|α �f(ξ))
+of a product of two or more functions, the fractional Leibniz rules, are used in the area of
+dispersive equations, see e.g. Kato–Ponce [15] and Grafakos–Oh [8]. We do not otherwise
+attempt to summarize the massive body of literature here and simply mention that the
+closest existing result is perhaps [18], which develops multiresolution methods in the
+non-entangled multilinear bi-parameter case.
+In this paper we prove the following “paraproduct free” T1 theorem for bilinear Zyg-
+mund SIOs.
+1.2. Theorem. Let T be a bilinear paraproduct free Calderón-Zygmund operator adapted to Zyg-
+mund dilations as in Definition 3.5. Let 1 < p1, p2 < ∞ and 1
+2 < p < ∞ with 1
+p :=
+1
+p1 + 1
+p2.
+Then we have
+∥T(f1, f2)∥Lp ≲ ∥f1∥Lp1∥f2∥Lp2.
+Notice that we can conclude the full bilinear range, including the quasi-Banach range,
+just from the paraproduct free T1 type assumptions. Also relevant is the fact that e.g. the
+appearing weak boundedness condition only involves Zygmund rectangles – that is, the
+T1 assumptions of Definition 3.5 are Zygmund adapted and in this respect weaker than
+the corresponding tri-parameter assumptions.
+It would also be very interesting to develop weighted theory with suitable kernel as-
+sumptions like in the linear case [14]. That is, to generalize our recent paper [19] from
+the standard multi-parameter setting to this entangled Zygmund setting. Recall that it
+would be key to deal with “genuine” multilinear weights, i.e., only impose a joint Ap
+condition on the associated tuple of weights ⃗w = (w1, . . . , wn). While such multilinear
+weighted estimates had been known for one-parameter SIOs for over 10 years by the
+influential paper [16], the multi-parameter version was only recently solved in [19]. The
+entangled situation is very difficult, though, and we do not achieve such estimates in
+this paper. Indeed, we are splitting our operators in a way that is sufficient for the un-
+bounded estimates, but not for the weighted estimates. In fact, already the unweighted
+estimates are surprisingly delicate and the only way we found to achieve them was with
+using this additional decomposition and even some sparse domination tools.
+Here is an outline of the paper. In Section 2 we develop the fundamental Zygmund
+adapted multiresolution methods in the bilinear setting. Section 3 introduces the sin-
+gular integrals and the corresponding testing conditions, and Section 4 uses the kernel
+estimates to bound the various coefficients arising in the multiresolution analysis. Sec-
+tion 5 contains a further decomposition of our dyadic model operators – this is then
+required in Section 6, where the Lp estimates of these model operators are proved. Sec-
+tion 6 concludes with the proof of Theorem 1.2. Section 7 contains the proof of the linear
+commutator estimates, Theorem 1.1, and the corresponding theory of product and lit-
+tle BMO commutators in the Zygmund setting. Appendix A considers bilinear variants
+of the multipliers studied by Fefferman-Pipher [6] – this is motivation for the abstract
+definitions of Section 3.
+2. BILINEAR ZYGMUND MULTIRESOLUTION ANALYSIS
+2.A. Dyadic intervals, Zygmund rectangles and basic randomization. Given a dyadic
+grid D, I ∈ D and k ∈ Z, k ≥ 0, we use the following notation:
+(1) ℓ(I) is the side length of I.
+
+4
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+(2) I(k) ∈ D is the kth parent of I, i.e., I ⊂ I(k) and ℓ(I(k)) = 2kℓ(I).
+(3) ch(I) is the collection of the children of I, i.e., ch(I) = {J ∈ D: J(1) = I}.
+(4) EIf = ⟨f⟩I1I is the averaging operator, where ⟨f⟩I =
+ffl
+I f =
+1
+|I|
+´
+I f.
+(5) ∆If is the martingale difference ∆If = �
+J∈ch(I) EJf − EIf.
+(6) ∆I,kf or ∆k
+If is the martingale difference block
+∆I,kf = ∆k
+If =
+�
+J∈D
+J(k)=I
+∆Jf.
+We will have use for randomization soon. While often the grids are fixed and we sup-
+press the dependence on the random parameters, it will be important to understand the
+definitions underneath. So we go ahead and introduce the related notation and standard
+results now. Let D0 be the standard dyadic grid in R. For ω ∈ {0, 1}Z, ω = (ωi)i∈Z, we
+define the shifted lattice
+D(ω) :=
+�
+L + ω := L +
+�
+i: 2−i<ℓ(L)
+2−iωi: L ∈ D0
+�
+.
+Let Pω be the product probability measure on {0, 1}Z. We recall the following notion of a
+good interval from [10]. We say that G ∈ D(ω, k), k ≥ 2, if G ∈ D(ω) and
+(2.1)
+d(G, ∂G(k)) ≥ ℓ(G(k))
+4
+= 2k−2ℓ(G).
+Notice that for all L ∈ D0 and k ≥ 2 we have
+(2.2)
+Pω({ω: L + ω ∈ D(ω, k)}) = 1
+2.
+The key implication (of practical use later) of G ∈ D(ω, k) is that for n ∈ Z with |n| ≤ 2k−2
+we have
+(2.3)
+(G ∔ n)(k) = G(k),
+G ∔ n := G + nℓ(G).
+In fact, we will not need much more of randomization – it only remains to move the
+notation to our actual setting of R3 = R × R2. We define for
+σ = (σ1, σ2, σ3) ∈ {0, 1}Z × {0, 1}Z × {0, 1}Z
+that
+D(σ) := D(σ1) × D(σ2) × D(σ3).
+Let
+Pσ := Pσ1 × Pσ2 × Pσ3.
+For k = (k1, k2, k3), k1, k2, k3 ≥ 2, we define
+D(σ, k) = D(σ1, k1) × D(σ2, k2) × D(σ3, k3).
+We also e.g. write
+D(σ, (k1, 0, k3)) = D(σ1, k1) × D(σ2) × D(σ3, k3),
+that is, a 0 will designate that we do not have goodness in that parameter.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+5
+As for most of the argument σ is fixed, it makes sense to mainly suppress it from the
+notation and abbreviate, whenever possible, that
+Dm = D(σm),
+D(σm, km) = Dm(km),
+m = 1, 2, 3.
+Then also
+D = D(σ) =
+3
+�
+m=1
+Dm,
+D(k) =
+3
+�
+m=1
+Dm(km).
+We define the Zygmund rectangles DZ ⊂ D by setting
+(2.4)
+DZ =
+�
+I =
+3
+�
+m=1
+Im ∈ D: ℓ(I1)ℓ(I2) = ℓ(I3)
+�
+.
+Obviously, DZ(k) is defined similarly as above but also requires �3
+m=1 Im ∈ D(k).
+2.B. Zygmund martingale differences. Given I = �3
+m=1 Im we define the Zygmund
+martingale difference operator
+∆I,Zf := ∆I1∆I2×I3f.
+2.5. Remark. We highlight that the martingale difference ∆I2×I3 is the one-parameter
+(and not the bi-parameter) martingale difference on the rectangle I2 × I3:
+∆I2×I3 = ∆I2∆I3 + EI2∆I3 + ∆I2EI3 ̸= ∆I2∆I3.
+Moreover, the above operators really act on the full product space but only on the given
+parameters – for instance, ∆I1f(x1, x2, x3) = ∆1
+I1f(x1, x2, x3) = (∆I1f(·, x2, x3))(x1).
+We recall the following facts from [14]. For a dyadic λ > 0 define the dilated lattices
+D2,3
+λ
+= {I2,3 ∈ D2,3 := D2 × D3 : ℓ(I3) = λℓ(I2)}.
+The basic Zygmund expansion goes as follows:
+f =
+�
+I1∈D1
+∆I1f =
+�
+I1∈D1
+�
+I2,3∈D2,3
+ℓ(I1)
+∆I1∆I2,3f =
+�
+I∈DZ
+∆I,Zf.
+(2.6)
+However, the way we split our operators will not be this simple.
+The following basic results hold for the martingale differences. For I, J ∈ DZ we have
+∆I,Z∆J,Zf =
+�
+∆I,Z
+if I = J,
+0
+if I ̸= J.
+Notice also that the Zygmund martingale differences satisfy
+ˆ
+R
+∆I,Zf dx1 = 0
+and
+ˆ
+R2 ∆I,Zf dx2 dx3 = 0.
+Moreover, we have
+ˆ
+(∆I,Zf)g =
+ˆ
+f∆I,Zg.
+
+6
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+2.C. Haar functions. For an interval J ⊂ R we denote by Jl and Jr the left and right
+halves of J, respectively. We define
+h0
+J = |J|−1/21J
+and
+h1
+J = hJ = |J|−1/2(1Jl − 1Jr).
+The reader should carefully notice that h0
+I is the non-cancellative Haar function for us
+and that in some other papers a different convention is used.
+As we mostly work on R3 = R × R2 we require some Haar functions on R2 as well.
+For I2 × I3 ⊂ R2 and η = (η2, η3) ∈ {0, 1}2 define
+hη
+I2×I3 = hη2
+I2 ⊗ hη3
+I3.
+Similarly, as hI1 denotes a cancellative Haar function on R, we let hI2×I3 denote a can-
+cellative one-parameter Haar function on I2 × I3. This means that
+hI2×I3 = hη
+I2×I3
+for some η = (η2, η3) ∈ {0, 1}2 \ {(0, 0)}. We only use a 0 to denote a non-cancellative
+Haar function: h0
+I2×I3 = h(0,0)
+I2×I3.
+We suppress this η dependence in all that follows in the sense that a finite η summation
+is not written. For example, given I = I1 × I2 × I3 ∈ DZ ⊂ �3
+m=1 Dm decompose
+∆I,Zf = ∆I1∆I2×I3f = ⟨f, hI1 ⊗ hI2×I3⟩hI1 ⊗ hI2×I3 =: ⟨f, hI,Z⟩hI,Z.
+2.D. Bilinear Zygmund shifts. In preparation for defining the shifts, we define the fol-
+lowing notation. Let I1, I2, I3 be rectangles, Ij = I1
+j × I2
+j × I3
+j = I1
+j × I2,3
+j , and f1, f2, f3 be
+functions defined on R3. For j1, j2 ∈ {1, 2, 3} define
+Aj1,j2
+I1,I2,I3 = Aj1,j2
+I1,I2,I3(f1, f2, f3) :=
+3
+�
+j=1
+⟨fj, vIj⟩,
+where
+vIj = �hI1
+j ⊗ �hI2,3
+j ;
+�hI1
+j1 = hI1
+j1
+and
+�hI1
+j = h0
+I1
+j , j ̸= j1;
+�hI2,3
+j2 = hI2,3
+j2
+and
+�hI2,3
+j
+= h0
+I2,3
+j , j ̸= j2.
+For a dyadic λ > 0 define
+Dλ = {K = K1 × K2 × K3 ∈ D: λℓ(K1)ℓ(K2) = ℓ(K3)}.
+Moreover, for a rectangle I = I1 × I2 × I3 and k = (k1, k2, k3) define
+I(k) = I(k1)
+1
+× I(k2)
+2
+× I(k3)
+3
+.
+2.7. Definition. Let k = (k1, k2, k3), ki ∈ {0, 1, 2, . . .}, be fixed. A bilinear Zygmund shift
+Q = Qk of complexity k has the form
+⟨Qk(f1, f2), f3⟩
+=
+�
+K∈D2−k1−k2+k3
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+aK,(Ij)
+�
+Aj1,j2
+I1,I2,I3 − Aj1,j2
+I1
+j1×I2,3
+1
+,I1
+j1×I2,3
+2
+,I1
+j1×I2,3
+3
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+7
+− Aj1,j2
+I1
+1×I2,3
+j2 ,I1
+2×I2,3
+j2 ,I1
+3×I2,3
+j2
++ Aj1,j2
+I1
+j1×I2,3
+j2 ,I1
+j1×I2,3
+j2 ,I1
+j1×I2,3
+j2
+�
+for some j1, j2 ∈ {1, 2, 3}. The coefficients aK,(Ij) satisfy
+|aK,(Ij)| ≤ |I1|1/2|I2|1/2|I3|1/2
+|K|2
+= |I1|3/2
+|K|2 .
+Now, the game is to represent bilinear singular integrals using the operators Qk and
+also – independently – bound the operators Qk suitably. We start with the representa-
+tion part and deal with bounding the operators later. We have not defined our singular
+integrals carefully yet, however, a lot of the required decomposition can be formally car-
+ried out for an arbitrary operator T. The singular integral part is later required to get
+sufficient decay for the appearing scalar coefficients and to handle the paraproducts.
+2.E. Zygmund decomposition of ⟨T(f1, f2), f3⟩. For now, we focus on the multireso-
+lution part and start formally decomposing a general bilinear operator. We begin by
+writing ⟨T(f1, f2), f3⟩ as
+�
+I1
+1,I1
+2,I1
+3∈D1
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
+=
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1),ℓ(I1
+2)>ℓ(I1
+3)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
++
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1),ℓ(I1
+3)>ℓ(I1
+2)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
++
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+2),ℓ(I1
+3)>ℓ(I1
+1)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
++
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1)>ℓ(I1
+2)=ℓ(I1
+3)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
++
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+2)>ℓ(I1
+1)=ℓ(I1
+3)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
++
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+3)>ℓ(I1
+1)=ℓ(I1
+2)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩
++
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1)=ℓ(I1
+2)=ℓ(I1
+3)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩.
+We collapse the first six sums, which are not already diagonal sums, into diagonal sums
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1)=ℓ(I1
+2)=ℓ(I1
+3)
+.
+
+8
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+This has the effect that whenever we have an inequality ℓ(I1
+i ) > ℓ(I1
+j ), the martingale
+difference operator ∆I1
+i corresponding with the larger cube is changed to the averaging
+operator EI1
+i . Thus, in the first three sums we now have two averaging operators, and in
+the next three we have one averaging operator. The more averaging operators we have,
+the less cancellation we have, and thus the main challenge are the first three sums with
+the least cancellation. We mainly focus on the first three sums for this reason.
+In addition, the first three sums are symmetric, so we may focus on only one of them,
+and choose to look at
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1),ℓ(I1
+2)>ℓ(I1
+3)
+⟨T(∆I1
+1f1, ∆I1
+2f2), ∆I1
+3f3⟩ =
+�
+I1
+1,I1
+2,I1
+3∈D1
+ℓ(I1
+1)=ℓ(I1
+2)=ℓ(I1
+3)
+⟨T(EI1
+1f1, EI1
+2f2), ∆I1
+3f3⟩.
+Now, we fix I1
+1, I1
+2, I1
+3 ∈ D1 with ℓ(I1
+1) = ℓ(I1
+2) = ℓ(I1
+3) and repeat the argument for
+⟨T(EI1
+1f1, EI1
+2f2), ∆I1
+3f3⟩ using the lattice D2,3
+ℓ(I1), where recall that for a dyadic λ > 0 we
+have
+D2,3
+λ
+= {I2 × I3 ∈ D2,3 := D2 × D3 : ℓ(I3) = λℓ(I2)}.
+This produces seven terms, and we again focus on
+�
+I2
+1×I3
+1,I2
+2×I3
+2,I2
+3×I3
+3∈D2,3
+ℓ(I1)
+ℓ(I2
+1)=ℓ(I2
+2)=ℓ(I2
+3)
+⟨T(EI1
+1EI2
+1×I3
+1f1, EI1
+2EI2
+2×I3
+2f2), ∆I1
+3∆I2
+3×I3
+3f3⟩.
+Altogether, our focus, for now, is on the key term
+(2.8)
+�
+I1,I2,I3∈DZ
+ℓ(I1)=ℓ(I2)=ℓ(I3)
+⟨T(EI1f1, EI2f2), ∆I3,Zf3⟩,
+where ℓ(I1) = ℓ(I2) = ℓ(I3) means that
+ℓ(Im
+1 ) = ℓ(Im
+2 ) = ℓ(Im
+3 ),
+m = 1, 2, 3.
+This was completely generic – we now go a step further to the direction of Zygmund
+shifts and start introducing Haar functions into the mix.
+2.F. Further decomposition of (2.8). Write
+⟨T(EI1f1, EI2f2), ∆I3,Zf3⟩ = ⟨T(h0
+I1, h0
+I2), hI3,Z⟩⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩⟨f3, hI3,Z⟩.
+Now, we perform a rather complicated decomposition of the product ⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩.
+To this end, start by writing
+⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩
+=
+�
+⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩ − ⟨f1, h0
+I1
+3h0
+I2,3
+1 ⟩⟨f2, h0
+I1
+3h0
+I2,3
+2 ⟩
+�
++ ⟨f1, h0
+I1
+3h0
+I2,3
+1 ⟩⟨f2, h0
+I1
+3 h0
+I2,3
+2 ⟩
+=: A1 + A2.
+We then further decompose A1 as follows
+A1 =
+�
+⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩ − ⟨f1, h0
+I1
+3h0
+I2,3
+1 ⟩⟨f2, h0
+I1
+3h0
+I2,3
+2 ⟩
+− ⟨f1, h0
+I1
+1h0
+I2,3
+3 ⟩⟨f2, h0
+I1
+2h0
+I2,3
+3 ⟩ + ⟨f1, h0
+I1
+3h0
+I2,3
+3 ⟩⟨f2, h0
+I1
+3 h0
+I2,3
+3 ⟩
+�
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+9
++
+�
+⟨f1, h0
+I1
+1 h0
+I2,3
+3 ⟩⟨f2, h0
+I1
+2 h0
+I2,3
+3 ⟩ − ⟨f1, h0
+I1
+3h0
+I2,3
+3 ⟩⟨f2, h0
+I1
+3h0
+I2,3
+3 ⟩
+�
+.
+When we later specialize to singular integrals, we will in particular make the following
+assumption. We say that T is a paraproduct free operator, if for all cancellative Haar
+functions hI1 and hI2,3 we have
+⟨T(1 ⊗ 1J2,3
+1 , 1 ⊗ 1J2,3
+2 ), hI1 ⊗ 1J2,3
+3 ⟩ = ⟨T ∗,j
+1 (1 ⊗ 1J2,3
+1 , 1 ⊗ 1J2,3
+2 ), hI1 ⊗ 1J2,3
+3 ⟩
+= ⟨T(1I1
+1 ⊗ 1, 1I1
+2 ⊗ 1), 1I1
+3 ⊗ hI2,3⟩ = ⟨T ∗,j
+2,3 (1I1
+1 ⊗ 1, 1I1
+2 ⊗ 1), 1I1
+3 ⊗ hI2,3⟩ = 0
+for all the adjoints j ∈ {1, 2}. With this assumption in the full summation (2.8) everything
+else vanishes except
+�
+I1,I2,I3∈DZ
+ℓ(I1)=ℓ(I2)=ℓ(I3)
+⟨T(h0
+I1, h0
+I2),hI3,Z⟩
+�
+⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩ − ⟨f1, h0
+I1
+3×I2,3
+1 ⟩⟨f2, h0
+I1
+3×I2,3
+2 ⟩
+− ⟨f1, h0
+I1
+1 ×I2,3
+3 ⟩⟨f2, h0
+I1
+2×I2,3
+3 ⟩ + ⟨f1, h0
+I3⟩⟨f2, h0
+I3⟩
+�
+⟨f3, hI3,Z⟩.
+So we eliminated the paraproducts by assumption, and now we have to manipulate this
+remaining term to a suitable form involving shifts.
+In the above sum we will relabel I3 = I = I1 × I2 × I3 = I1 × I2,3. Then, for n1 =
+(n1
+1, n2
+1, n3
+1) = (n1
+1, n2,3
+1 ) we write
+I1 = I ∔ n1 = (I1 + n1
+1ℓ(I1)) × (I2 + n2
+1ℓ(I2)) × (I3 + n3
+1ℓ(I3)) = (I1 ∔ n1
+1) × (I2,3 ∔ n2,3
+1 ).
+We write I2 similarly as I2 = I ∔ n2. Notice that if n1
+1 = n1
+2 = 0, then the term inside
+the summation vanishes. Similarly, if n2,3
+1
+= n2,3
+2
+= (0, 0), the term inside the summation
+vanishes. So we need to study
+�
+n1,n2∈Z3
+max(|n1
+1|,|n1
+2|)̸=0
+max(|n2
+1|,|n2
+2|)̸=0 or max(|n3
+1|,|n3
+2|)̸=0
+�
+I∈DZ
+cI,n1,n2,
+where
+cI,n1,n2
+= ⟨T(h0
+I∔n1, h0
+I∔n2), hI,Z⟩
+�
+⟨f1, h0
+I∔n1⟩⟨f2, h0
+I∔n2⟩ − ⟨f1, h0
+I1×(I2,3∔n2,3
+1 )⟩⟨f2, h0
+I1×(I2,3∔n2,3
+2
+)⟩
+− ⟨f1, h0
+(I1∔n1
+1)×I2,3⟩⟨f2, h0
+(I1∔n1
+2)×I2,3⟩ + ⟨f1, h0
+I⟩⟨f2, h0
+I⟩
+�
+⟨f3, hI,Z⟩.
+We write
+�
+n1,n2∈Z3
+max
+j=1,2 |n1
+j|̸=0
+max
+j=1,2 |n2
+j|̸=0 or max
+j=1,2 |n3
+j|̸=0
+�
+I∈DZ
+cI,n1,n2
+=
+∞
+�
+k1,k2,k3=2
+�
+n1,n2∈Z3
+max
+j=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2,3
+�
+I∈DZ
+cI,n1,n2
+
+10
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
++
+∞
+�
+k1,k2=2
+�
+n1,n2∈Z3
+max
+j=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2
+n3
+1=n3
+2=0
+�
+I∈DZ
+cI,n1,n2
++ Σsym,
+where Σsym is symmetric to the second term and has n2
+1 = n2
+2 = 0.
+Recall how everything implicitly depends on the random parameter σ, so that we can
+average over it. By independence, we have by (2.2) that
+Eσ
+∞
+�
+k1,k2,k3=2
+�
+n1,n2∈Z3
+max
+j=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2,3
+�
+I∈DZ
+cI,n1,n2
+= 8Eσ
+∞
+�
+k1,k2,k3=2
+�
+n1,n2∈Z3
+max
+j=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2,3
+�
+I∈DZ(k)
+cI,n1,n2,
+k = (k1, k2, k3).
+(2.9)
+For the other two terms, where n2
+j = 0 or n3
+j = 0, we perform the above but do not add
+goodness to the second and third parameters, respectively. For example, we have
+Eσ
+∞
+�
+k1,k2=2
+�
+n1,n2∈Z3
+max
+j=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2
+n3
+1=n3
+2=0
+�
+I∈DZ
+cI,n1,n2
+= 4Eσ
+∞
+�
+k1,k2=2
+�
+n1,n2∈Z3
+max
+j=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2
+n3
+1=n3
+2=0
+�
+I∈DZ(k1,k2,0)
+cI,n1,n2.
+Continuing with (2.9), we write it as
+C8Eσ
+∞
+�
+k1,k2,k3=2
+(|k| + 1)2ϕ(k)
+�
+K∈Dλ
+�
+I∈DZ(k)
+I(k)=K
+�
+n1,n2∈Z3
+maxj=1,2 |nm
+j |∈(2km−3,2km−2]
+m=1,2,3
+cI,n1,n2
+C(|k| + 1)2ϕ(k),
+where
+Dλ = {K = K1 × K2 × K3 ∈ D: λℓ(K1)ℓ(K2) = ℓ(K3)},
+λ = 2k3−k1−k2,
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+11
+and C is some suitably large constant depending on T. Recall that by (2.3) we also have
+(I ∔ n1)(k) = (I ∔ n2)(k) = I(k) = K.
+We have arrived to a point where we cannot go further without talking about singular
+integrals. Indeed, we need kernel estimates to control the coefficients. But on a structural
+level (with the paraproduct free assumption), we have obtained a reasonable representa-
+tion of the main term (2.8) in terms of sums of bilinear Zygmund shifts.
+3. BILINEAR ZYGMUND SINGULAR INTEGRALS
+We begin by defining the required kernel estimates and cancellation conditions for
+bilinear singular integrals T invariant under Zygmund dilations. For motivation for the
+form of the kernel estimates, see Appendix A for kernel bounds of bilinear multipliers.
+This viewpoint makes the kernel estimates natural – on the other hand, they are also of
+the right form so that we will be able to bound the coef���cients from the multiresolution
+decomposition and obtain reasonable decay.
+3.A. Full kernel representation. Our bilinear singular integral T invariant under Zyg-
+mund dilations is related to a full kernel K in the following way. The kernel K is a
+function
+K : (R3 × R3 × R3) \ ∆ → C,
+where
+∆ = {(x, y, z) ∈ R3 × R3 × R3 : xi = yi = zi for at least one i = 1, 2, 3}.
+We look at the action of T on rectangles like I1 × I2 × I3 =: I1 × I2,3 in R3 = R × R × R =
+R × R2. So let Ii = I1
+i × I2
+i × I3
+i be rectangles, i = 1, 2, 3. Assume that there exists
+i1, i2, j1, j2 ∈ {1, 2, 3} so that I1
+i1 and I1
+i2 are disjoint and also I2,3
+j1
+and I2,3
+j2
+are disjoint.
+Then we have the full kernel representation
+⟨T(1I1, 1I2), 1I3⟩ =
+˚
+K(x, y, z)1I1(x)1I2(y)1I3(z) dx dy dz.
+The kernel K satisfies the following estimates.
+First, we define the decay factor
+Dθ(x, y) =
+��2
+i=1(|xi| + |yi|)
+|x3| + |y3|
++
+|x3| + |y3|
+�2
+i=1(|xi| + |yi|)
+�−θ
+,
+θ ∈ (0, 2],
+and the tri-parameter bilinear size factor
+S(x, y) =
+3
+�
+i=1
+1
+�
+|xi| + |yi|
+�2 .
+We demand the following size estimate
+(3.1)
+|K(x, y, z)| ≲ Dθ(x − z, y − z)S(x − z, y − z).
+
+12
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+Let now c = (c1, c2, c3) be such that |ci − xi| ≤ max(|xi − zi|, |yi − zi|)/2 for i = 1, 2, 3.
+We assume that K satisfies the mixed size and Hölder estimates
+|K((c1,x2, x3), y, z) − K(x, y, z)|
+≲
+�
+|c1 − x1|
+|x1 − z1| + |y1 − z1|
+�α1Dθ(x − z, y − z)S(x − z, y − z),
+(3.2)
+and
+|K((x1, c2, c3), y, z) − K(x, y, z)|
+≲
+�
+|c2 − x2|
+|x2 − z2| + |y2 − z2| +
+|c3 − x3|
+|x3 − z3| + |y3 − z3|
+�α23Dθ(x − z, y − z)S(x − z, y − z),
+(3.3)
+where α1, α23 ∈ (0, 1]. Finally, we assume that K satisfies the Hölder estimate
+|K(c, y, z) − K((c1, x2, x3), y, z) − K((x1, c2, c3), y, z) + K(x, y, z)|
+≲
+�
+|c1 − x1|
+|x1 − z1| + |y1 − z1|
+�α1�
+|c2 − x2|
+|x2 − z2| + |y2 − z2| +
+|c3 − x3|
+|x3 − z3| + |y3 − z3|
+�α23
+× Dθ(x − z, y − z)S(x − z, y − z).
+(3.4)
+We also demand the symmetrical mixed size and Hölder estimates and Hölder estimates.
+For j = 1, 2, define the adjoint kernels K∗,j, K∗,j
+1
+and K∗,j
+2,3 via the natural formulas, e.g.,
+K∗,1(x, y, z) = K(z, y, x),
+K∗,2
+1 (x, y, z) = K(x, (z1, y2, y3), (y1, z2, z3)).
+We assume that each adjoint kernel satisfies the same estimates as the kernel K.
+3.B. Partial kernel representations. Let �θ ∈ (0, 1]. For every interval I1 we assume that
+there exists a kernel
+KI1 : (R2 × R2 × R2) \ {(x2,3, y2,3, z2,3): xi = yi = zi for i = 2 or i = 3} → C,
+so that if I2,3
+j1 and I2,3
+j2 are disjoint for some j1, j2 ∈ {1, 2, 3}, then
+⟨T(1I1 ⊗ 1I2,3
+1 , 1I1 ⊗ 1I2,3
+2 ), 1I1 ⊗ 1I2,3
+3 ⟩
+=
+˚
+KI1(x2,3, y2,3, z2,3)1I2,3
+1 (x2,3)1I2,3
+2 (y2,3)1I2,3
+3 (z2,3) dx2,3 dy2,3 dz2,3.
+We demand the following estimates for the kernel KI1 : The size estimate
+|KI1(x2,3, y2,3, z2,3)|
+≲
+�|I1|(|x2 − z2| + |y2 − z2|)
+|x3 − z3| + |y3 − z3|
++
+|x3 − z3| + |y3 − z3|
+|I1|(|x2 − z2| + |y2 − z2|)
+�−�θ
+|I1|
+�3
+i=2
+�
+|xi − zi| + |yi − zi|
+�2
+and the continuity estimate
+|KI1(c2,3, y2,3, z2,3) − KI1(x2,3, y2,3, z2,3)|
+≲
+�
+|c2 − x2|
+|x2 − z2| + |y2 − z2| +
+|c3 − x3|
+|x3 − z3| + |y3 − z3|
+�α23
+×
+�|I1|(|x2 − z2| + |y2 − z2|)
+|x3 − z3| + |y3 − z3|
++
+|x3 − z3| + |y3 − z3|
+|I1|(|x2 − z2| + |y2 − z2|)
+�−�θ
+|I1|
+�3
+i=2
+�
+|xi − zi| + |yi − zi|
+�2
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+13
+whenever c2,3 = (c2, c3) is such that |ci − xi| ≤ max(|xi − zi|, |yi − zi|)/2 for i = 2, 3. We
+also assume the symmetrical continuity estimates.
+We assume similar one-parameter conditions for the other partial kernel representa-
+tion. That is, for every rectangle I2,3, there exists a standard bilinear Calderón-Zygmund
+kernel KI2,3 so that if I1
+j1 and I1
+j2 are disjoint for some j1, j2 ∈ {1, 2, 3}, then
+⟨T(1I1
+1 ⊗ 1I2,3, 1I1
+2 ⊗ 1I2,3), 1I1
+3 ⊗ 1I2,3⟩
+=
+˚
+KI2,3(x1, y1, z1)1I1
+1 (x1)1I1
+2 (y1)1I1
+3(z1) dx1 dy1 dz1.
+The kernel KI2,3 satisfies the standard estimates
+|KI2,3(x1, y1, z1)| ≤ CKI2,3
+1
+(|x1 − z1| + |y1 − z1|)2 ,
+|KI2,3(x1, y1, z1) − KI2,3(c1, y1, z1)| ≤ CKI2,3
+|x1 − c1|α1
+(|x1 − z1| + |y1 − z1|)2+α1
+whenever |x1 − c1| ≤ max(|x1 − z1|, |y1 − z1|)/2, and the symmetric continuity estimates.
+The smallest possible constant CKI2,3 in these inequalities is denoted by ∥KI2,3∥CZα1. We
+then assume that
+∥KI2,3∥CZα1 ≲ |I2,3|.
+3.C. Cancellation assumptions: paraproduct free operators. We say that T is a para-
+product free operator, if for all cancellative Haar functions hI1 and hI2,3 we have
+⟨T(1 ⊗ 1J2,3
+1 , 1 ⊗ 1J2,3
+2 ), hI1 ⊗ 1J2,3
+3 ⟩ = ⟨T ∗,j
+1 (1 ⊗ 1J2,3
+1 , 1 ⊗ 1J2,3
+2 ), hI1 ⊗ 1J2,3
+3 ⟩
+= ⟨T(1I1
+1 ⊗ 1, 1I1
+2 ⊗ 1), 1I1
+3 ⊗ hI2,3⟩ = ⟨T ��,j
+2,3 (1I1
+1 ⊗ 1, 1I1
+2 ⊗ 1), 1I1
+3 ⊗ hI2,3⟩ = 0
+for all the adjoints j ∈ {1, 2}. We always assume that all bilinear Zygmund operators
+in this article satisfy this cancellation condition. The intention of this condition is to
+guarantee that our operator is representable using cancellative shifts only.
+3.D. Weak boundedness property. We say that T satisfies the weak boundedness prop-
+erty if
+|⟨T(1I, 1I), 1I⟩| ≲ |I|
+for all Zygmund rectangles I = I1 × I2 × I3.
+3.5. Definition. We say that a bilinear operator T is a paraproduct free Calderón-
+Zygmund operator adapted to Zygmund dilations (CZZ operator) if T has the full kernel
+representation, the partial kernel representations, is paraproduct free and satisfies the
+weak boundedness property.
+4. ESTIMATES FOR THE SHIFT COEFFICIENTS
+We now move to consider the shift coefficients that appeared in the decomposition
+of T in Section 2.F. When T is a CZZ operator, we can estimate them. Without loss of
+generality, we estimate
+⟨T(h0
+I ˙+n1, h0
+I ˙+n2), hI,Z⟩
+for I ∈ DZ and different values of n1, n2 ∈ Z3, and without loss of generality we assume
+θ = ˜θ < 1. The coefficients related to the other terms of the decomposition (other than the
+main term (2.8)) may have a different set of Haar functions, but they are treated similarly.
+
+14
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+We show that
+(4.1)
+|⟨T(h0
+I ˙+n1, h0
+I ˙+n2), hI,Z⟩| ≲ (|k| + 1)2ϕ(k) |I|
+3
+2
+|K|2 ,
+where
+ϕ(k) := 2−k1α1−k2 min{α23,θ}−max{k3−k1−k2,0}θ.
+For terms of this particular form, we would not actually need to analyze some of the
+diagonal cases (see Section 2.F). However, these diagonal terms would appear in some
+other forms, so it makes sense to deal with them here (even though in the real situation
+the Haar functions might be permuted differently, this does not matter, and the calcula-
+tions we present apply). It is very helpful to study the linear case [14], since the kernel
+estimates are relatively involved and we will not repeat every detail when they are simi-
+lar.
+Let mi := maxj=1,2 |ni
+j|. The analysis of the coefficients splits to combinations of
+
+
+
+
+
+|m1| ∈ (2k1−3, 2k1−2],
+k1 = 3, 4, . . . ,
+(Separated)
+|m1| = 1,
+(Adjacent)
+|m1| = 0,
+(Identical)
+and
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+|mi| ∈ (2ki−3, 2ki−2],
+i = 2, 3, ki = 3, 4, . . . ,
+(Separated)
+|m2| < 2 and |m3| ∈ (2k3−3, 2k3−2],
+k3 = 3, 4, . . . ,
+(Separated)
+|m2| ∈ (2k2−3, 2k2−2] and |m3| < 2
+k2 = 3, 4, . . . ,
+(Separated)
+|m2| = 1 and |m3| ≤ 1
+(Adjacent)
+|m2| = 0 and |m3| = 1
+(Adjacent)
+m2 = 0 = m3.
+(Identical)
+It is enough to consider mi = ni
+1 since the case mi = ni
+2 is symmetrical. We will not go
+through explicitly every combination – rather, we choose some illustrative examples.
+4..1. Separated/Separated. We begin with the case |ni
+1| ≥ 2 for all i = 1, 2, 3. Hence,
+|xi − zi| ≥ |ni
+1|ℓ(Ii) ≥ 2ki−3ℓ(Ii)
+and
+|xi − zi| ≤ |ni
+1|ℓ(Ii) + 2ℓ(Ii) ≤ 2ki−1ℓ(Ii)
+for i = 1, 2, 3. Moreover, |xi − zi| ≥ |yi − zi|/2 ≥ 0 for i = 1, 2, 3. Thus, we have the
+estimate
+��2
+i=1(|xi − zi| + |yi − zi|)
+(|x3 − z3| + |y3 − z3|)
++
+|x3 − z3| + |y3 − z3|
+�2
+i=1(|xi − zi| + |yi − zi|)
+�−θ
+∼
+��2
+i=1 |xi − zi|
+|x3 − z3|
++
+|x3 − z3|
+�2
+i=1 |xi − zi|
+�−θ
+∼
+��2
+i=1 2kiℓ(Ii)
+2k3ℓ(I3)
++
+2k3ℓ(I3)
+�2
+i=1 2kiℓ(Ii)
+�−θ
+= (2k1+k2−k3 + 2k3−k1−k2)−θ.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+15
+Using the cancellation of the Haar function we then have
+���
+˚
+K(x, y, z)h0
+I ˙+n1(x)h0
+I ˙+n2(y)hI,Z(z) dx dy dz
+���
+=
+���
+˚ �
+K(x, y, z) − K(x, y, (cI1, z2,3)) − K(x, y, (z1, cI2,3)) + K(x, y, cI)
+�
+× h0
+I ˙+n1(x)h0
+I ˙+n2(y)hI,Z(z) dx dy dz
+���
+≲
+˚
+2−k1α1(2−k2 + 2−k3)α23 (2k1+k2−k3 + 2k3−k1−k2)−θ
+|K|2
+h0
+I ˙+n1(x)h0
+I ˙+n2(y)h0
+I(z) dx dy dz
+= 2−k1α1(2−k2 + 2−k3)α23(2k1+k2−k3 + 2k3−k1−k2)−θ |I|
+3
+2
+|K|2 ≤ ϕ(k) |I|
+3
+2
+|K|2 .
+Let us then consider the case, where we have separation in the parameter 3 but not in
+the parameter 2 – that is, |n2
+1| < 2 ≤ |n3
+1|. Then
+��2
+i=1(|xi − zi| + |yi − zi|)
+|x3 − z3| + |y3 − z3|
++
+|x3 − z3| + |y3 − z3|
+�2
+i=1(|xi − zi| + |yi − zi|)
+�−θ
+(4.2)
+∼
+�|x2 − z2| + |y2 − z2|
+2k3−k1|I2|
++
+2k3−k1|I2|
+|x2 − z2| + |y2 − z2|
+�−θ
+≲
+� |x2 − z2|
+2k3−k1|I2| + 2k3−k1|I2|
+|x2 − z2|
+�−θ
++
+� |y2 − z2|
+2k3−k1|I2| + 2k3−k1|I2|
+|y2 − z2|
+�−θ
+,
+and so using the mixed estimates
+���
+˚
+K(x, y, z)h0
+I ˙+n1(x)h0
+I ˙+n2(y)hI,Z(z) dx dy dz
+���
+=
+���
+˚ �
+K(x, y, z) − K(x, y, (cI1, z2,3))
+�
+h0
+I ˙+n1(x)h0
+I ˙+n2(y)hI,Z(z) dx dy dz
+���
+≲
+˚
+2−k1α1|K1|−2|K3|−2
+�
+|x2−z2|+|y2−z2|
+2k3−k1|I2|
++
+2k3−k1|I2|
+|x2−z2|+|y2−z2|
+�−θ
+�
+|x2 − z2| + |y2 − z2|
+�2
+× h0
+I ˙+n1(x)h0
+I ˙+n2(y)h0
+I(z) dx dy dz
+= 2−k1α1 |I1|
+3
+2|I3|
+3
+2
+|K1|2|K3|2
+˚
+�
+|x2−z2|+|y2−z2|
+2k3−k1|I2|
++
+2k3−k1|I2|
+|x2−z2|+|y2−z2|
+�−θ
+�
+|x2 − z2| + |y2 − z2|
+�2
+× h0
+I2 ˙+n2
+1(x2)h0
+I2 ˙+n2
+2(y2)h0
+I2(z2) dx2 dy2 dz2
+≲ ϕ(k) |I|
+3
+2
+|K|2 .
+We note that the last inequality requires a case study (see also [14, Lemma 8.5]) and we
+used the standard estimate
+ˆ
+Rd
+du
+(r + |u0 − u|)d+α ≲ r−α.
+(4.3)
+Symmetrical estimates hold if |n2
+1| ≥ 2 > |n3
+1|.
+
+16
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+4..2. Adjacent/Separated. We look at the example case |n2
+1| ≥ 2 > |n3
+1| and |n1
+1| = 1. By the
+size estimate we have
+|⟨T(h0
+I ˙+n1, h0
+I ˙+n2), hI,Z⟩|
+≲
+|I2|3/2
+|I1,3|3/2|K2|2
+¨
+�
+(|x1−z1|+|y1−z1|)2k2ℓ(I2)
+|x3−z3|+|y3−z3|
++
+|x3−z3|+|y3−z3|
+(|x1−z1|+|y1−z1|)2k2ℓ(I2)
+�−θ
+�
+|x1 − z1| + |y1 − z1|
+�2�
+|x3 − z3| + |y3 − z3|
+�2
+× 1I1,3 ˙+n1,3
+1 (x1,3)1I1,3 ˙+n1,3
+2 (y1,3)1I1,3(z1,3) dx1,3 dy1,3 dz1,3.
+Similarly as (4.2), we can split the integral into two terms. Then by (4.3) we reduce the
+problem to estimating
+¨
+�
+(|x1−z1|+|y1−z1|)2k2ℓ(I2)
+|x3−z3|
++
+|x3−z3|
+(|x1−z1|+|y1−z1|)2k2ℓ(I2)
+�−θ
+�
+|x1 − z1| + |y1 − z1|
+�2|x3 − z3|
+× 1I1,3 ˙+n1,3
+1 (x1,3)1I1 ˙+n1
+2(y1)1I1,3(z1,3) dx1,3 dy1 dz1,3
++
+¨
+�
+(|x1−z1|+|y1−z1|)2k2ℓ(I2)
+|y3−z3|
++
+|y3−z3|
+�
+|x1−z1|+|y1−z1|
+�
+2k2ℓ(I2)
+�−θ
+(|x1 − z1| + |y1 − z1|)2|y3 − z3|
+× 1I1,3 ˙+n1,3
+1 (x1,3)1I1,3 ˙+n1,3
+2 (y1,3)1I1(z1) dx1,3 dy1,3 dz1.
+Since they are similar, we only bound the first one. Note that
+�(|x1 − z1| + |y1 − z1|)2k2ℓ(I2)
+|x3 − z3|
++
+|x3 − z3|
+(|x1 − z1| + |y1 − z1|)2k2ℓ(I2)
+�−θ
+× (|x1 − z1| + |y1 − z1|)−2
+≤
+�(|x1 − z1| + |y1 − z1|)2k2ℓ(I2)
+|x3 − z3|
+�−θ
+(|x1 − z1| + |y1 − z1|)−2χ{|x1−z1|2k2ℓ(I2)≥|x3−z3|}
++
+�
+|x3 − z3|
+(|x1 − z1| + |y1 − z1|)2k2ℓ(I2)
+�−θ
+(|x1 − z1| + |y1 − z1|)−2χ{|x1−z1|2k2ℓ(I2)<|x3−z3|}.
+Then apply (4.3) to the integral over y1, then by following the linear case [14, Lemma
+8.11] we get that the above integral is bounded by |I1,3|k22−k2θ. Thus, we get
+|⟨T(h0
+I ˙+n1, h0
+I ˙+n2), hI,Z⟩| ≲
+|I2|3/2
+|I1,3|1/2|K2|2 k22−k2θ ≲ k2ϕ(k) |I|
+3
+2
+|K|2 .
+4..3. Adjacent/Adjacent. We again have no major changes to the linear case but in order
+to use the estimate
+(4.4)
+ˆ
+R
+�
+t
+|u| + |u|
+t
+�−θ
+t|u|
+|f(u)| du ≲ t−1Mf(0)
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+17
+we need to first use (4.3) repeatedly. For example, consider |n1
+1| = 1 and |n2
+1| = 1, |n3
+1| ≤ 1.
+By the size estimate of the kernel, we need to control
+� �2
+i=1(|xi−zi|+|yi−zi|)
+|x3−z3|+|y3−z3|
++
+|x3−z3|+|y3−z3|
+�2
+i=1(|xi−zi|+|yi−zi|)
+�−θ
+�3
+i=1
+�
+|xi − zi| + |yi − zi|
+�2
+h0
+I ˙+n1(x)h0
+I ˙+n2(y)h0
+I(z).
+As before, we split this into two terms, one of them is
+� �2
+i=1(|xi−zi|+|yi−zi|)
+|x3−z3|
++
+|x3−z3|
+�2
+i=1(|xi−zi|+|yi−zi|)
+�−θ
+�3
+i=1
+�
+|xi − zi| + |yi − zi|
+�2
+h0
+I ˙+n1(x)h0
+I ˙+n2(y)h0
+I(z).
+We then apply (4.3) to the integral over y3, and then use the previous trick repeatedly.
+That is, we write
+��2
+i=1(|xi − zi| + |yi − zi|)
+|x3 − z3|
++
+|x3 − z3|
+�2
+i=1(|xi − zi| + |yi − zi|)
+�−θ
+≤
+��2
+i=1(|xi − zi| + |yi − zi|)
+|x3 − z3|
+�−θ
+χ{|x1−z1|(|x2−z2|+|y2−z2|)≥|x3−z3|}
++
+�
+|x3 − z3|
+�2
+i=1(|xi − zi| + |yi − zi|)
+�−θ
+χ{|x1−z1|(|x2−z2|+|y2−z2|)<|x3−z3|}
+and apply (4.3) to the integral over y1. Then, after a similar argument on y2, we finally
+arrive at
+1
+|I|
+1
+2
+¨
+� �2
+i=1 |xi−zi|
+|x3−z3|
++
+|x3−z3|
+�2
+i=1 |xi−zi|
+�−θ
+�3
+i=1 |xi − zi|
+h0
+I ˙+n1(x)h0
+I(z) dx dz
+≲
+1
+|I|
+1
+2
+≲ |I|
+3
+2
+|K|2 .
+4..4. Adjacent/Identical. We consider the case |n1
+1| = 1 and n2
+j = n3
+j = 0, j = 1, 2. We write
+�
+Q2,3
+1
+,Q2,3
+2 ,Q2,3
+3 ∈ch(I2,3)
+⟨T(h0
+I ˙+n11Q2,3
+1 , h0
+I ˙+n21Q2,3
+2 ), hI,Z1Q2,3
+3 ⟩.
+It is enough to consider Q2,3
+1
+= Q2,3
+2
+= Q2,3
+3
+since otherwise we have adjacent intervals,
+and we are back in the Adjacent/Adjacent case. Hence, the partial kernel representation
+3.B yields that
+��� ± |I2,3|− 3
+2
+˚
+KQ2,3
+1 h0
+I1 ˙+n1
+1h0
+I1 ˙+n1
+2hI1
+���
+≲ |I2,3|
+3
+2
+|K2,3|2
+˚
+1
+(|x1 − z1| + |y1 − z1|)2 h0
+I1 ˙+n1
+1(x1)h0
+I1 ˙+n1
+2(y1)hI1(z1) dx1 dy1 dz1.
+Then, first using (4.3) and then standard integration methods we get the following in-
+equality
+˚
+1
+(|x1 − z1| + |y1 − z1|)2 h0
+I1 ˙+n1
+1(x1)h0
+I1 ˙+n1
+2(y1)hI1(z1) dx1 dy1 dz1
+
+18
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+≲
+1
+|I1|
+1
+2
+¨
+1
+|x1 − z1|h0
+I1 ˙+n1
+1(x1)hI1(z1) dx1 dz1
+��
+1
+|I1|
+1
+2
+∼ |I1|
+3
+2
+|K1|2
+as desired.
+4..5. Identical/Identical. Just like in above we split the pairing to
+�
+Q1
+1,Q1
+2,Q1
+3∈ch(I1)
+�
+Q2,3
+1
+,Q2,3
+2 ,Q2,3
+3 ∈ch(I2,3)
+⟨T(h0
+I ˙+n1(1Q1
+1 ⊗ 1Q2,3
+1 ), h0
+I ˙+n2(1Q1
+2 ⊗ 1Q2,3
+2 )), hI,Z(1Q1
+3 ⊗ 1Q2,3
+3 )⟩.
+The cases when Q1
+i ̸= Q1
+j for some i, j = 1, 2, 3, i ̸= j are essentially included in the cases
+of the two previous subsections. Hence, we consider Q1
+1 = Q1
+2 = Q1
+3. Then there are two
+cases left, that is, either Q2,3
+i
+̸= Q2,3
+j
+for some i, j = 1, 2, 3, i ̸= j, or Q2,3
+1
+= Q2,3
+2
+= Q2,3
+3 .
+Beginning from the latter one, we directly see that
+|⟨T(1Q1
+1 ⊗ 1Q2,3
+1 , 1Q1
+1 ⊗ 1Q2,3
+1 ), 1Q1
+1 ⊗ 1Q2,3
+1 ⟩| ≲ |Q1
+1||Q2,3
+1 |
+by the weak boundedness property 3.D. Hence, we get the desired bound
+|⟨T(h0
+I ˙+n1(1Q1
+1 ⊗ 1Q2,3
+1 ), h0
+I ˙+n2(1Q1
+1 ⊗ 1Q2,3
+1 )), hI,Z(1Q1
+1 ⊗ 1Q2,3
+1 )⟩| ≲ |Q1|
+|I|
+3
+2
+≤ |I|
+3
+2
+|K|2 .
+We handle the remaining case Q2,3
+i
+̸= Q2,3
+j
+for some i, j = 1, 2, 3, i ̸= j. By the partial
+kernel representation and its size estimate we get
+��� ± |I|− 3
+2
+˚
+KQ1
+11Q2,3
+1 1Q2,3
+2 1Q2,3
+3
+���
+≲
+1
+|I1|
+1
+2
+1
+|I2,3|
+3
+2
+˚ �|I1|(|x2 − z2| + |y2 − z2|)
+|x3 − z3| + |y3 − z3|
++
+|x3 − z3| + |y3 − z3|
+|I1|(|x2 − z2| + |y2 − z2|)
+�−θ
+×
+3
+�
+i=2
+1
+�
+|xi − zi| + |yi − zi|
+�2 1Q2,3
+1 1Q2,3
+2 1Q2,3
+3 dx2,3 dy2,3 dz2,3.
+Then using similar arguments as in the Adjacent/Adjacent case and (4.4) gives us the
+desired bound.
+5. STRUCTURAL DECOMPOSITION OF ZYGMUND SHIFTS
+In this section we decompose the bilinear Zygmund shifts (see Section 2.D) as a sum
+of operators with simpler cancellation properties. The decomposition is not optimal (in
+the sense that weighted estimates with Zygmund weights cannot be obtained via this) –
+however, it is sufficient for unweighted boundedness in the full range that we later obtain
+via tri-parameter theory. Recall that k = (k1, k2, k3) is the complexity of the bilinear
+Zygmund shift.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+19
+5.1. Definition. Bilinear operators of the form
+(5.2)
+S(l1,l2,l3)(f1, f2) =
+�
+L∈Dλ
+�
+I
+(ℓj)
+j
+=L
+aL,(Ij)⟨f1, hI1
+1 ⊗ h0
+I2,3
+1 ⟩⟨f2, h0
+I1
+2 ⊗ hI2,3
+2 ⟩hI3,
+where λ = 2n, n ∈ Z, |n| ≤ 3 max(ki) and
+|aL,(Ij)| ≤ |Ij|
+3
+2
+|L|2 ,
+are tri-parameter bilinear shifts of Zygmund nature if at least one rectangle I1
+i1 ×I2,3
+i2 , i1 =
+1, 3, i2 = 2, 3 is a Zygmund rectangle and
+(1) ℓi
+j ≤ ki for all i, j = 1, 2, 3;
+(2) (ℓ3
+j − ℓ2
+j)+ ≤ (k3 − k2)+ for all j = 1, 2, 3.
+Moreover, any adjoint
+S
+j∗
+1,j∗
+2,3
+(l1,l2,l3),
+j1, j2,3 ∈ {0, 1, 2},
+is also considered to be a tri-parameter bilinear shift of Zygmund nature. Here, the ad-
+joint j∗
+2,3 means that, for example, in case j2,3 = 1 functions h0
+I2,3
+1
+and hI2,3
+3
+switch places.
+Note that these operators share a ‘weaker’ Zygmund structure. Ideally, we would
+want to have I3 ∈ DZ and I1
+1 × I2,3
+2
+∈ DZ.
+5.3. Proposition. Let Qk, k = (k1, k2, k3), be a bilinear Zygmund shift operator as defined in
+Section 2.D. Then
+Qk = C
+c
+�
+u=1
+k1−1
+�
+l1=0
+k2,3−1
+�
+l2,3=0
+Su,
+where Su is a bilinear operator as in Definition 5.1 with complexity depending on l and k,and
+k2,3−1
+�
+l2,3=0
+:=
+
+
+
+
+
+
+
+
+
+
+
+�
+0≤l2=l3≤k2−1
++
+�
+l2=k2
+k2≤l3≤k3−1
+,
+if k3 ≥ k2
+�
+0≤l2=l3≤k3−1
++
+�
+k3≤l2≤k2−1
+l3=k3
+,
+if k3 < k2.
+Proof. The argument is similar in spirit to the purely bi-parameter decomposition in [1].
+For notational convenience, we consider a shift Qk of the particular form
+⟨Qk(f1, f2), f3⟩
+=
+�
+K∈D2−k1−k2+k3
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+aK,(Ij)
+�
+A3,3
+I1,I2,I3 − A3,3
+I1
+3×I2,3
+1
+,I1
+3×I2,3
+2
+,I3
+− A3,3
+I1
+1×I2,3
+3
+,I1
+2×I2,3
+3
+,I3 + A3,3
+I3,I3,I3
+�
+=
+�
+K∈D2−k1−k2+k3
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+aK,(Ij)⟨f3, hI3⟩
+�
+⟨f1, h0
+I1⟩⟨f2, h0
+I2⟩ − ⟨f1, h0
+I1
+3h0
+I2,3
+1 ⟩⟨f2, h0
+I1
+3h0
+I2,3
+2 ⟩
+
+20
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+− ⟨f1, h0
+I1
+1h0
+I2,3
+3 ⟩⟨f2, h0
+I1
+2h0
+I2,3
+3 ⟩ + ⟨f1, h0
+I3⟩⟨f2, h0
+I3⟩
+�
+.
+There is no essential difference in the general case. Let us also use the usual abbreviation
+D2−k1−k2+k3 = Dλ.
+We define
+bK,(Ij) = |I1|aK,(Ij)
+and
+B3,3
+I1,I2,I3 = ⟨f1⟩I1⟨f2⟩I2⟨f3, hI3⟩.
+We can write the shift Qk using these by replacing a with b and A with B.
+Recall the notation
+∆l1
+K1f =
+�
+L1∈D1
+(L1)(l1)=K1
+∆L1f,
+P k1
+K1f =
+k1−1
+�
+l1=0
+∆l1
+K1f,
+EK1f = ⟨f⟩K11K1,
+Ek1
+K1f =
+�
+L1∈D1
+(L1)(k1)=K1
+⟨f⟩L11L1.
+Let us define
+(5.4)
+P k2,3
+K2,3f :=
+k2,3−1
+�
+l2,3=0
+∆(l2,l3)
+K2,3 f :=
+
+
+
+
+
+
+
+
+
+k2−1
+�
+l2=0
+∆l2
+K2,3f +
+k3−1
+�
+l3=k2 Ek2
+K2∆l3
+K3f,
+if k3 ≥ k2
+k3−1
+���
+l3=0
+∆l3
+K2,3f +
+k2−1
+�
+l2=k3 ∆l2
+K2Ek3
+K3f,
+if k3 < k2,
+where we have the standard one-parameter definition
+∆li
+K2,3f =
+�
+L2,3∈D2,3
+(L2)(li)×(L3)(li)=K2×K3
+∆L2,3f.
+We also use a similar shorthand for the expanded martingale blocks
+k2,3−1
+�
+l2,3=0
+∆(l2,l3)
+K2,3 f =
+k2,3−1
+�
+l2,3=0
+�
+(L2,3)(l2,3)=K2,3
+⟨f, hL2,3⟩hL2,3,
+where we allow, for example, that hL2,3 = h0
+L2 ⊗ hL3 when k3 > k2 and l2 = k2.
+Using this notation we define the following. For a cube I and integers l, j0 ∈ {1, 2, . . . }
+we define
+(5.5)
+DI,l(j, j0) =
+
+
+
+
+
+EI,
+if j ∈ {1, . . . , j0 − 1},
+P l
+I,
+if j = j0,
+id,
+if j ∈ {j0 + 1, j0 + 2, . . . },
+where id denotes the identity operator, and if we have a rectangle I2,3 and a tuple l2,3 we
+use the modified P l2,3
+I2,3.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+21
+Let I1, I2, I3 be as in the summation of Qk. We use the above notation in parameter one
+DI1,l1(j, j0) and for the other two parameters we use DI2,3,l2,3(j, j0). Thus, expanding to
+the martingale blocks leads us to
+B3,3
+I1,I2,I3
+=
+3
+�
+m1,m2=1
+2
+�
+j=1
+⟨D1
+K1,k1(j, m1)D2,3
+K2,3,k2,3(j, m2)fj⟩Ij⟨f3, hI3⟩.
+Hence, we may write
+�
+K∈Dλ
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+B3,3
+I1,I2,I3 =:
+3
+�
+m1,m2=1
+Σ1
+m1,m2.
+Also, we have that
+B3,3
+I1
+3×I2,3
+1
+,I1
+3×I2,3
+2
+,I1
+3×I2,3
+3
+=
+3
+�
+m2=1
+2
+�
+j=1
+⟨D2,3
+K2,3,k2,3(j, m2)fj⟩I1
+3×I2,3
+j ⟨f3, hI3⟩
+and
+B3,3
+I1
+1×I2,3
+3
+,I1
+2×I2,3
+3
+,I1
+3×I2,3
+3
+=
+3
+�
+m1=1
+2
+�
+j=1
+⟨D1
+K1,k1(j, m1)fj⟩I1
+j ×I2,3
+3 ⟨f3, hI3⟩,
+which gives that
+�
+K∈Dλ
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+B3,3
+I1
+3×I2,3
+1
+,I1
+3×I2,3
+2
+,I1
+3×I2,3
+3
+=:
+3
+�
+m2=1
+Σ2
+m2
+and
+�
+K∈Dλ
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+B3,3
+I1
+1×I2,3
+3
+,I1
+2×I2,3
+3
+,I1
+3×I2,3
+3
+=:
+3
+�
+m1=1
+Σ3
+m1.
+Finally, we just set
+�
+K∈Dλ
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+B3,3
+I3,I3,I3 =: Σ4.
+Thus, we have the following decomposition
+⟨Qk(f1, f2), f3⟩ =
+2
+�
+m1,m2=1
+Σ1
+m1,m2 +
+2
+�
+m2=1
+(Σ1
+3,m2 − Σ2
+m2)
++
+2
+�
+m1=1
+(Σ1
+m1,3 − Σ3
+m1) + (Σ1
+3,3 − Σ2
+3 − Σ3
+3 + Σ4).
+
+22
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+First, we take one Σ1
+m1,m2 with m1, m2 ∈ {1, 2}. For notational convenience, we choose
+the case m1 = m2 = 2. Recall that
+Σ1
+2,2 =
+�
+K∈Dλ
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+bK,(Ij)⟨f1⟩K⟨P k1
+K1P k2,3
+K2,3f2⟩I2⟨f3, hI3⟩.
+We expand
+⟨P k1
+K1P k2,3
+K2,3f2⟩I2 =
+k1−1
+�
+l1=0
+k2,3−1
+�
+l2,3=0
+�
+(L1)(l1)=K1
+(L2,3)(l2,3)=K2,3
+⟨f2, hL1 ⊗ hL2,3⟩⟨hL1 ⊗ hL2,3⟩I2
+and note that L is not necessarily a Zygmund rectangle. It holds that
+Σ1
+2,2 =
+k1−1
+�
+l1=0
+k2,3−1
+�
+l2,3=0
+�
+K∈Dλ
+�
+L(l1,l2,l3)=K
+�
+I3∈DZ
+I(k)
+3
+=K
+� �
+I1
+I(k)
+1
+=K
+�
+I2⊂L
+I(k)
+2
+=K
+bK,(Ij)⟨hL⟩I2
+|K|
+1
+2
+�
+⟨f1, h0
+K⟩⟨f2, hL⟩⟨f3, hI3⟩.
+Now, since we can easily check that
+���
+�
+I1
+I(k)
+1
+=K
+�
+I2⊂L
+I(k)
+2
+=K
+bK,(Ij)⟨hL⟩I2
+|K|
+1
+2
+��� ≤ |K|
+1
+2 |L|
+1
+2 |I3|
+1
+2
+|K|2
+,
+we get a sum of operators we wanted
+Σ1
+2,2 =
+k1−1
+�
+l1=0
+k2,3−1
+�
+l2,3=0
+⟨S(0,(l1,l2,l3),k)(f1, f2), f3⟩,
+where S(0,(l1,l2,l3),k) is a type of the shift (5.2). The general case Σ1
+m1,m2 is analogous.
+We turn to the terms Σ1
+3,m2 − Σ2
+m2. Let us take, for example, the case m2 = 1. After
+expanding P k2,3
+K2,3 in the first slot, Σ1
+3,1 − Σ2
+1 can be written as
+k2,3−1
+�
+l2,3=0
+�
+K∈Dλ
+�
+(L2,3)(l2,3)=K2,3
+�
+I1,I2,I3
+I(k)
+j
+=K
+bK,(Ij)⟨hL2,3⟩I2,3
+1
+��
+f1, 1K1
+|K1| ⊗ hL2,3
+�
+⟨f2⟩K1×I2,3
+2
+−
+�
+f1,
+1I1
+3
+|I1
+3| ⊗ hL2,3
+�
+⟨f2⟩I1
+3×I2,3
+2
+�
+⟨f3, hI3⟩.
+For the moment, we fix one l2,3 and write g1 = ⟨f1, hL2⟩ and g2 = ⟨f2⟩I2,3
+2 . We write inside
+the brackets
+2
+�
+j=1
+⟨gj⟩K1 −
+2
+�
+j=1
+⟨gj⟩I1
+3 = −
+k1−1
+�
+l1=0
+�
+2
+�
+j=1
+⟨gj⟩(I1
+3)(l1) −
+2
+�
+j=1
+⟨gj⟩(I1
+3)(l1+1)
+�
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+23
+and then expand �2
+j=1⟨gj⟩(I1
+3)(l1) − �2
+j=1⟨gj⟩(I1
+3)(l1+1) as
+⟨∆(I1
+3)(l1+1)g1⟩I1
+3⟨g2⟩(I1
+3)(l1) + ⟨g1⟩(I1
+3)(l1+1)⟨∆(I1
+3)(l1+1)g2⟩I1
+3.
+We get
+2
+�
+j=1
+⟨gj⟩K1 −
+2
+�
+j=1
+⟨gj⟩I1
+3
+= −
+k1−1
+�
+l1=0
+�
+⟨∆(I1
+3)(l1+1)g1⟩I1
+3⟨g2⟩(I1
+3)(l1) + ⟨g1⟩(I1
+3)(l1+1)⟨∆(I1
+3)(l1+1)g2⟩I1
+3
+�
+,
+where we can expand
+⟨∆(I1
+3)(l1+1)gj⟩I1
+3 = ⟨gj, h(I1
+3)(l1+1)⟩⟨h(I1
+3 )(l1+1)⟩I1
+3.
+For fixed l1 and l2,3 the expansion leads to the term
+�
+K∈Dλ
+�
+(L2,3)(l2,3)=K2,3
+�
+I1,I2,I3
+I(k)
+j
+=K
+bK,(Ij)⟨h(I1
+3)(l1+1) ⊗ hL2,3⟩I1
+3×I2,3
+1
+�
+f1, h(I1
+3 )(l1+1) ⊗ hL2,3
+�
+⟨f2⟩(I1
+3)(l1)×I2,3
+2 ⟨f3, hI3⟩,
+and to the symmetrical one, where the cancellation h(I1
+3)(l1+1) is paired with the second
+function and f1 is averaged over (I1
+3)(l1+1). Again, we want to reorganize the summations
+and verify the correct normalization for the shifts of the form (5.2). In the first parameter
+we will now take (I1
+3)(l1+1) as the new top cube, that is,
+�
+K1
+�
+(L1)(k1−l1)=K1
+�
+K2,3∈D2−l1−k2+k3 ℓ(L1)
+�
+(I1
+3)(l1)=L1
+�
+(L2,3)(l2,3)=K2,3
+�
+I2,3
+2
+,I2,3
+3
+(Ii
+j)(ki)=Ki
+cK1,L1,I1
+3,K2,3,L2,3,I2,3
+2
+,I2,3
+3
+�
+f1, h(L1)(1) ⊗ hL2,3
+�
+⟨f2⟩L1×I2,3
+2 ⟨f3, hI3⟩,
+(5.6)
+where
+cK1,L1,I1
+3,K2,3,L2,3,I2,3
+2
+,I2,3
+3
+=
+�
+I1
+1,I1
+2
+(I1
+j )(k1)=K1
+�
+I2,3
+1
+⊂L2,3
+(Ii
+1)(ki)=Ki
+bK,(Ij)⟨h(L1)(1)×L2,3⟩I1
+3×I2,3
+1 .
+Moreover, we have
+|cK1,L1,I1
+3,K2,3,L2,3,I2,3
+2
+,I2,3
+3 | ≤ |(L1)(1)|
+3
+2|I1
+3|
+1
+2
+|(L1)(1)|2
+× |L2,3|
+1
+2|I2,3
+2 ||I2,3
+3 |
+1
+2
+|K2,3|2
+.
+Notice that this is the right normalization for (5.2), since f2 is related to L1 and |(L1)(1)| =
+2|L1|, and we can change the averages into pairings against non-cancellative Haar func-
+tions.
+
+24
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+We conclude that for some C ≥ 1 we have
+C−1(5.6) = ⟨S((0,l2,3),(1,k2,3),(l1+1,k2,3))(f1, f2), f3⟩,
+where S((0,l2,3),(1,k2,3),(l1+1,k2,3)) is an operator of the desired type and of complexity
+(0, l2,3), (1, k2,3), (l1 + 1, k2,3).
+The other term and the other case of Σ1
+3,2 − Σ2
+2 are analogous.
+The cases Σ1
+m1,3 − Σ3
+m1 are handled almost identically, however, we need to treat
+2
+�
+j=1
+⟨gj⟩K2,3 −
+2
+�
+j=1
+⟨gj⟩I2,3
+3
+slightly differently. We expand the rectangles I2,3
+3
+in the one-parameter fashion until we
+reach the smaller of the cubes K2, K3. Then we continue with one-parameter expansion
+with only one of the cubes until we reach the bigger of the cubes K2, K3. For example, if
+k3 > k2, we expand as
+2
+�
+j=1
+⟨gj⟩K2,3 −
+2
+�
+j=1
+⟨gj⟩I2,3
+3
+= −
+k2−1
+�
+l2=0
+�
+⟨∆(I2,3
+3
+)(l2+1,l2+1)g1⟩(I2,3
+3
+)(l2,l2)⟨g2⟩(I2,3
+3
+)(l2,l2)
++ ⟨g1⟩(I2,3
+3
+)(l2+1,l2+1)⟨∆(I2,3
+3
+)(l2+1,l2+1)g2⟩(I2,3
+3
+)(l2,l2)
+�
+−
+k3−1
+�
+l3=k2
+�
+⟨EK2∆(I3
+3)(l3+1)g1⟩K2×(I3
+3)(l3)⟨g2⟩K2×(I3
+3)(l3)
++ ⟨g1⟩K2×(I3
+3)(l3+1)⟨EK2∆(I3)(l3+1)g2⟩K2×(I3
+3)(l3)
+�
+,
+The case k3 ≤ k2 can be expanded similarly. Similarly as in the previous cases, we can
+now write the terms in the particular form (5.2). For example, related to the latter term,
+�
+K∈Dλ
+�
+L2,3∈D2,3
+λl,kℓ(K1)
+L2=K2
+(L3)(k3−l3)=K3
+�
+(L1)(l1)=K1
+�
+(I1
+3)(k1)=K1
+�
+(I2
+3)(k2)=K2
+(I3
+3)(l3)=L3
+cK,L,I3
+�
+f1, 1K1
+|K1| ⊗ h(L2,3)(0,1)
+��
+f2, hL1 ⊗ 1L2,3
+|L2,3|
+�
+⟨f3, hI3⟩,
+where l3 ∈ {k2, . . . , k3 − 1}, λl,k = 2−k1−k2+l3 and
+|cK,L,I3| =
+���
+�
+I1,I2
+(Ij)(k)=K
+I1
+2⊂L1
+aK,(Ij)|I1|⟨hL1 ⊗ hK2×(L3)(1)⟩I1
+2×K2×L3
+���
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+25
+≤
+�
+I1,I2
+(Ij)(k)=K
+I1
+2⊂L1
+|I3|
+1
+2|I1||I2|
+|K|2
+|K2|− 1
+2|(L3)(1)|− 1
+2 |L1|− 1
+2
+= |L1|
+1
+2
+|K1|
+|I3|
+1
+2|K2 × (L3)(1)|
+3
+2
+|K2 × (L3)(1)|2
+.
+This normalization is an absolute constant away from the correct one since we consider
+that K2 × (L3)(1) is the top rectangle in parameters 2 and 3.
+Finally, we consider Σ1
+3,3 − Σ2
+3 − Σ3
+3 + Σ4 that equals to
+�
+K∈Dλ
+�
+I1,I2,I3∈DZ
+I(k)
+j
+=K
+bK,(Ij)
+�
+2
+�
+j=1
+⟨fj⟩K −
+2
+�
+j=1
+⟨fj⟩I1
+3×K2,3 −
+2
+�
+j=1
+⟨fj⟩K1×I2,3
+3
++
+2
+�
+j=1
+⟨fj⟩I3
+�
+⟨f3, hI3⟩.
+(5.7)
+As we already showed, we can expand
+2
+�
+j=1
+⟨fj⟩K −
+2
+�
+j=1
+⟨fj⟩I1
+3×K2,3
+= −
+k1−1
+�
+l1=0
+�
+⟨∆(I1
+3)(l1+1)g1⟩I1
+3⟨g2⟩(I1
+3)(l1) + ⟨g1⟩(I1
+3)(l1+1)⟨∆(I1
+3)(l1+1)g2⟩I1
+3
+�
+,
+where gj = ⟨fj⟩K2,3, and similarly for
+n
+�
+j=1
+⟨fj⟩I3 −
+2
+�
+j=1
+⟨fj⟩K1×I2,3
+3
+we get same expansion with the positive sign and gj = ⟨fj⟩I2,3
+3 .
+Then we sum the two expansions together and expand in the parameters 2 and 3. That
+is, we will expand
+k1−1
+�
+l1=0
+⟨h(I1
+3 )(l1+1)⟩(I1
+3)(l1)
+�
+f1, h(I1
+3 )(l1+1) ⊗ 1K2,3
+|K2,3|
+�
+⟨f2⟩(I1
+3 )(l1)×K2,3
+−
+�
+f1, h(I1
+3 )(l1+1) ⊗
+1I2,3
+3
+|I2,3
+3 |
+�
+⟨f2⟩(I1
+3)(l1)×I2,3
+3 .
+Thus, we get, for example when k2 < k3, that
+k1−1
+�
+l1=0
+k2−1
+�
+l2=0
+�
+K∈Dλ
+�
+L1∈D1
+(L1)(k1−l1)=K1
+�
+L2,3∈D2,3
+2−l1 ℓ(L1)
+(L2,3)(k2−l2,k3−l2)=K2,3
+�
+I3∈DZ
+(I3)(l1,l2,l2)=L
+× cK,L,I3
+�
+f1, h(L1)(1) ⊗ h0
+(L2,3)(1,1)
+��
+f2, h0
+L1 ⊗ h(L2,3)(1,1)
+�
+⟨f3, hI3⟩
+
+26
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
++
+k1−1
+�
+l1=0
+k3−1
+�
+l3=k2
+�
+K∈Dλ
+�
+L1∈D1
+(L1)(k1−l1)=K1
+�
+L2,3∈D2−l1−k2+l3 ℓ(L1)
+L2=K2
+(L3)(k3−l3)=K3
+�
+I3∈DZ
+(I3)(l1,k2,l3)=L
+× cK,L,I3
+�
+f1, h(L1)(1) ⊗ h0
+(L2,3)(0,1)
+��
+f2, h0
+L1 ⊗ h(L2,3)(0,1)
+�
+⟨f3, hI3⟩.
+Here
+|cK,L,I3| =
+���
+�
+I1,I2∈DZ
+Ik
+j =K
+aK,(Ij)|I1||L1|− 1
+2 |(L2,3)(1)|− 1
+2 ⟨h(L1)(1) ⊗ h(L2,3)(1)⟩L1×L2,3
+���
+≤ |I3|
+1
+2|(L1)(1)|
+3
+2
+|(L)(1)|2
+|L1|− 1
+2 |(L2,3)(1)|− 1
+2 ∼ |I3|
+1
+2 |(L1)(1)|
+1
+2 |L1|
+1
+2|(L2,3)(1)|
+3
+2
+|(L1)(1)|2|(L2,3)(1)|2
+.
+We abused notation slightly by (L2,3)(1) meaning both (L2,3)(1,1) and (L2,3)(0,1). The other
+terms are handled analogously.
+□
+6. BOUNDEDNESS OF ZYGMUND SHIFTS
+In this section we prove the boundedness of Zygmund shifts. We first prove the fol-
+lowing. A collection S is called γ-sparse if there are pairwise disjoint subsets E(S) ⊂ S,
+S ∈ S , with |E(S)| ≥ γ|S|. Often the precise value of γ is not important and we just talk
+about sparse collections.
+6.1. Proposition. Let λ = 2k for some k ∈ Z and
+Λ(f1, f2, f3) =
+�
+K∈D2,3
+λ
+�
+(Ij)(ℓj )=K
+�3
+j=1 |Ij|
+1
+2
+|K|2
+|⟨f1, h0
+I1⟩| · |⟨f2, hI2⟩| · |⟨f3, hI3⟩|.
+Then there exists a sparse collection S ⊂ D2,3
+λ
+such that
+Λ(f1, f2, f3) ≲ max{k2, k3}
+�
+S∈S
+|S|
+3
+�
+j=1
+⟨|fj|⟩S.
+Proof. The proof is an easy adaptation of the sparseness argument in [17, Section 5]. In
+fact, we only need to check the validity of
+Λ(f1, f2, f3) ≲ ∥f1∥Lp∥f2∥Lq∥f3∥Lr,
+where p, q, r ∈ (1, ∞) and 1/p + 1/q + 1/r = 1. This can be done by direct computation:
+Λ(f1, f2, f3) ≤
+ˆ
+f1
+�
+K∈D2,3
+λ
+⟨|∆ℓ2
+Kf2|⟩K⟨|∆ℓ3
+Kf3|⟩K1K
+≤ ∥f1∥Lp
+���
+�
+�
+K∈D2,3
+λ
+�
+MD2,3
+λ |∆ℓ2
+Kf2|
+�2� 1
+2 ���
+Lq
+���
+�
+�
+K∈D2,3
+λ
+�
+MD2,3
+λ |∆ℓ3
+Kf3|
+�2� 1
+2 ���
+Lr
+≲ ∥f1∥Lp∥f2∥Lq∥f3∥Lr.
+□
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+27
+6.2. Proposition. Let Qk, k = (k1, k2, k3), be a bilinear Zygmund shift as in Section 2.D, and
+let 1 < p1, p2 < ∞ and 1
+2 < p < ∞ with 1
+p :=
+1
+p1 + 1
+p2. Let
+w1, w2 ∈ Ap(R × R × R),
+and
+w := w
+p
+p1
+1 w
+p
+p2
+2 .
+Then, for every η ∈ (0, 1) we have
+∥Qk(f1, f2)∥Lp(w) ≲ max
+i {ki}22k1η∥f1∥Lp1(w1)∥f2∥Lp2(w2).
+Proof. We prove the weighted boundedness L4(w1)×L4(w2) → L2(w), of the tri-parameter
+bilinear shifts of Zygmund nature (5.2). We do this with tri-parameter weights wi ∈ A4.
+We then extrapolate the result to the full bilinear range using the traditional multilinear
+extrapolation by Grafakos–Martell (and Duoandikoetxea) [4,7]. Our result then follows
+from Proposition 5.3.
+Note that if we have I3 ∈ DZ in (5.2), then the related λ in Proposition 6.1 is
+2ℓ3
+3−ℓ2
+3−ℓ1
+3|L1|.
+(For other cases, for instance if I1
+1 × I2,3
+2
+∈ DZ, then λ = 2ℓ3
+2−ℓ2
+2−ℓ1
+1|L1|). Assume v ∈
+A4,λ(R2); recall that Ap,λ(R2) is defined similarly as Ap(R2) except that the supremum is
+taken over rectangles R = I × J with |J| = λ|I|. Then
+�
+S∈S
+|S|
+3
+�
+j=1
+⟨|fj|⟩S =
+�
+S∈S
+⟨|f1|⟩S⟨|f2|⟩S⟨|f3|v−1⟩v
+Sv(S).
+Since for any R ∈ S,
+�
+S⊂R
+S∈S
+v(S) =
+�
+S⊂R
+S∈S
+v(S)
+|S| |S| ≲
+�
+S⊂R
+S∈S
+v(S)
+|S| |ES| ≤
+ˆ
+R
+MD2,3
+λ (v1R) ≲[v]A4,λ(R2) v(R),
+by the Carleson embedding theorem we have
+(6.3)
+�
+S∈S
+|S|
+3
+�
+j=1
+⟨|fj|⟩S ≲[v]A4,λ(R2)
+ˆ
+R2 MD2,3
+λ |f1|MD2,3
+λ |f2|Mv
+D2,3
+λ (|f3|v−1)v.
+Now, given weights wj ∈ A4(R3), j = 1, 2, we know that w = w1/2
+1
+w1/2
+2
+∈ A4(R3). We
+have
+|⟨S(f1, f2), f3⟩| =
+�
+L1
+�
+(I1
+j )
+(ℓ1
+j )=L1
+�3
+j=1 |I1
+j |
+1
+2
+|L1|2
+Λ(⟨f1, hI1
+1⟩, ⟨f2, h0
+I1
+2⟩, ⟨f3, hI1
+3 ⟩).
+Note that ⟨w⟩L1 ∈ A4,λ(R2) with [⟨w⟩L1]A4,λ(R2) ≤ [w]A4 for any λ. Thus, applying (6.3)
+with v = ⟨w⟩L1 we have
+|⟨S(f1, f2), f3⟩|
+≲ max
+i
+{ki}
+�
+L1
+�
+(I1
+j )
+(ℓ1
+j )=L1
+�3
+j=1 |I1
+j |
+1
+2
+|L1|2
+ˆ
+R2 MD2,3
+λ ⟨f1, hI1
+1⟩MD2,3
+λ ⟨f2, h0
+I1
+2⟩Mv
+D2,3
+λ (⟨f3, hI1
+3⟩v−1)v
+
+28
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+= max
+i
+{ki}
+�
+L1
+ˆ
+R3⟨MD|∆ℓ1
+1
+L1f1|⟩L1⟨MD|f2|⟩L1
+�
+(I1
+3)(ℓ1
+3)=L1
+|I1
+3|
+1
+2 M
+⟨w⟩L1
+D2,3
+λ
+(⟨f3, hI1
+3⟩⟨w⟩−1
+L1 ) 1L1
+|L1|w
+≤ max
+i
+{ki}
+���
+� �
+L1
+�
+MD1MD|∆ℓ1
+1
+L1f1|
+�2� 1
+2 ���
+L4(w1)∥MD1MD|f2|∥L4(w2)
+×
+���
+� �
+L1
+�
+�
+(I1
+3)(ℓ1
+3)=L1
+|I1
+3|
+1
+2 M
+⟨w⟩L1
+D2,3
+λ
+(⟨f3, hI1
+3⟩⟨w⟩−1
+L1 )|L1|−1�2
+1L1
+� 1
+2 ���
+L2(w).
+By the well-know square function and maximal function estimates we have
+���
+� �
+L1
+�
+MD1MD|∆ℓ1
+1
+L1f1|
+�2� 1
+2 ���
+L4(w1) ≲ ∥f1∥L4(w1)
+and
+∥MD1MD|f2|∥L4(w2) ≲ ∥f2∥L4(w2).
+The estimate of the last term is a bit tricky. By the (one parameter)vector-valued estimates
+of M
+⟨w⟩L1
+D2,3
+λ
+(see e.g. [19, Proposition 4.3] for a bi-parameter version (the proof easily adapts
+to the one-parameter case)), we have
+���
+� �
+L1
+�
+�
+(I1
+3)(ℓ1
+3)=L1
+|I1
+3|
+1
+2M
+⟨w⟩L1
+D2,3
+λ
+(⟨f3, hI1
+3⟩⟨w⟩−1
+L1 )|L1|−1�2
+1L1
+� 1
+2 ���
+L2(w)
+≤ 2ℓ1
+3�����
+� �
+L1
+�
+�
+(I1
+3)(ℓ1
+3)=L1
+|I1
+3|
+s
+2M
+⟨w⟩L1
+D2,3
+λ
+(⟨f3, hI1
+3⟩⟨w⟩−1
+L1 )s|L1|− s
+2
+� 2
+s � 1
+2���
+L2(⟨w⟩L1)
+≲ 2ℓ1
+3��
+� �
+L1
+�
+�
+(I1
+3)(ℓ1
+3)=L1
+|I1
+3|
+s
+2��⟨f3, hI1
+3⟩⟨w⟩−1
+L1
+��s|L1|− s
+2
+� 2
+s � 1
+2 ���
+L2(⟨w⟩L1)
+≤ 2ℓ1
+3��
+� �
+L1
+�
+�
+(I1
+3)(ℓ1
+3)=L1
+|I1
+3|
+1
+2|⟨f3, hI1
+3⟩|⟨w⟩−1
+L1 |L1|− 1
+2
+�2� 1
+2 ���
+L2(⟨w⟩L1)
+≲ 2ℓ1
+3η∥f3∥L2(w−1),
+where s = (1/η)′ and in the last step we have used [19, Proposition 5.8]. Thus,
+∥S(f1, f2)∥L2(w) ≲ max
+i
+{ki}2k1η∥f1∥L4(w1)∥f2∥L4(w2).
+□
+Now we are able to conclude the proof of Theorem 1.2.
+Proof of Theorem 1.2. By the representation formula discussed in Sections 2.E and 2.F, the
+coefficient estimates in Section 4 (in particular (4.1)) we get that
+⟨T(f1, f2), f3⟩ =CEσ
+∞
+�
+k1,k2,k3=2
+(|k| + 1)2ϕ(k)
+�
+I∈DZ(k)
+⟨Q(k1,k2,k3)(f1, f2), f3⟩
+C(|k| + 1)2ϕ(k)
+.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+29
+Thus, for p1, p2 ∈ (1, ∞) so that p ∈ (1, ∞), we conclude by Proposition 6.2 that
+∥T(f1, f2)∥Lp(w) ≲
+∞
+�
+k1,k2,k3=2
+(|k| + 1)2ϕ(k) max
+i {ki}22k1η∥f1∥Lp1(w1)∥f2∥Lp2(w2)
+≲ ∥f1∥Lp1(w1)∥f2∥Lp2(w2),
+where we need to take η < α1. Consequently, we can now pass the result to the full
+bilinear range using the traditional multilinear extrapolation [4,7].
+□
+7. LINEAR COMMUTATORS IN THE ZYGMUND DILATION SETTING
+In this section we return to the linear theory and complete the following commutator
+estimate left open by previous results. This requires new and interesting paraproduct
+estimates. For the context, see the explanation below.
+7.1. Theorem. Let b ∈ L1
+loc and T be a linear CZZ operator as in [14]. Let θ ∈ (0, 1] be the
+kernel exponent measuring the decay in terms of the Zygmund ratio
+Dθ(x) :=
+�|x1x2|
+|x3|
++
+|x3|
+|x1x2|
+�−θ
+.
+Then
+∥[b, T]∥Lp→Lp ≲ ∥b∥bmoZ
+whenever p ∈ (1, ∞).
+Here the definition of the little BMO is given by
+∥b∥bmoZ := sup
+DZ
+sup
+R∈DZ
+1
+|R|
+ˆ
+R
+|b(x) − ⟨b⟩R| dx < ∞,
+where the supremum is over all different collections of Zygmund rectangles DZ and then
+over all R ∈ DZ.
+This theorem was previously considered in [5] using the so-called Cauchy trick. That
+method requires weighted bounds with Zygmund weights. But we now know [14] how
+delicate such weighted bounds are – weighted bounds with Zygmund weights do not
+in general hold if θ < 1. However, the commutator bounds are still true – but we need
+a different proof, presented here. It suffices to prove the boundedness of commutators
+[b, Qk] for any linear shift Qk of the Zygmund dilation type.
+For θ = 1 we could use the Cauchy trick and the weighted bounds from [14] – this
+would give weighted commutator estimates with Zygmund weights.
+We begin by recording lemmas that we need for the main proofs of this section.
+7.2. Lemma. Let b be a locally integrable function. Then the following are equivalent
+(1) b ∈ bmoDZ,
+(2)
+max
+�
+sup
+I1∈D1 ∥⟨b⟩I1,1∥BMOD2,3
+ℓ(I1)
+, ess sup
+(x2,x3)∈R2 ∥b(·, x2, x3)∥BMO
+�
+< ∞,
+(3)
+max
+�
+sup
+I2∈D2 ∥⟨b⟩I2,2∥BMOD2,3
+ℓ(I2)
+, ess sup
+(x1,x3)∈R2 ∥b(x1, ·, x3)∥BMO
+�
+< ∞.
+
+30
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+For completeness, we give the proof.
+Proof. Let us begin showing that bmoZ =⇒ (2) (and by symmetry also (3)). Clearly, for
+all Zygmund rectangles I = I1 × I2 × I3 ∈ DZ we have
+∥b∥bmoZ ≥ 1
+|I|
+ˆ
+I
+|b − ⟨b⟩I| ≥
+1
+|I2,3|
+ˆ
+I2,3 |⟨b⟩I1,1 − ⟨b⟩I|.
+(7.3)
+So by uniform boundedness we immediately get
+∥⟨b⟩I1,1∥BMOD2,3
+ℓ(I1)
+:=
+sup
+I2,3∈D2,3
+ℓ(I1)
+1
+|I2,3|
+ˆ
+I2,3 |⟨b⟩I1,1 − ⟨⟨b⟩I1,1⟩I2,3| ≤ ∥b∥bmoZ < ∞.
+We move on to proving the second assertion inside (2). For fixed I1 ∈ D1 we define
+fI1(x2, x3) :=
+´
+I1 |b(x1, x2, x3) − ⟨b⟩I1(x2, x3)| dx1. Then for every I2,3 ∈ D2,3
+ℓ(I1) we have
+⟨fI1⟩I2,3 ≤
+1
+|I2,3|
+ˆ
+I2,3
+ˆ
+I1 |b − ⟨b⟩I| +
+1
+|I2,3|
+ˆ
+I2,3
+ˆ
+I1 |⟨b⟩I1,1 − ⟨b⟩I| ≤ 2|I1|∥b∥bmoZ,
+where last inequality holds by definition and the above estimate (7.3). Now, by the
+Lebesgue differentiation theorem we get for (x2, x3) ∈ R2 \ N(I1), where N(I1) is a
+null set depending on I1, that
+fI1(x2, x3) ≤ 2|I1|∥b∥bmoZ.
+It is then easy to conclude that
+∥b(·, x2, x3)∥BMO ≤ 2∥b∥bmoZ
+for almost every (x2, x3) ∈ R2.
+Conversely,
+ˆ
+I
+|b − ⟨b⟩I| ≤
+ˆ
+I
+|b − ⟨b⟩I1,1| +
+ˆ
+I
+|⟨b⟩I1,1 − ⟨b⟩I|
+≤ |I1|
+ˆ
+I2,3 ∥b(·, x2, x3)∥BMO + |I|∥⟨b⟩I1,1∥BMOℓ(I1) ≤ |I|(C1 + C2),
+where C1 := ess sup(x2,x3)∈R2 ∥b(·, x2, x3)∥BMO and C2 := supI1 ∥⟨b⟩I1,1∥BMOℓ(I1).
+□
+Then the usual duality results imply the following.
+7.4. Corollary. If b ∈ bmoZ and I1 is fixed, then
+�
+I2,3∈D2,3
+ℓ(I1)
+⟨⟨b⟩I1, hI2,3⟩ϕI2,3 ≲ ∥b∥bmoZ
+���
+�
+�
+I2,3∈D2,3
+ℓ(I1)
+ϕI2,3 1I2,3
+|I2,3|
+� 1
+2���
+L1.
+Also, for fixed (x2, x3), we have
+�
+I1∈D1
+⟨b, hI1⟩1ϕI1 ≲ ∥b∥bmoZ
+���
+� �
+I1∈D1
+ϕI1 1I1
+|I1|
+� 1
+2���
+L1.
+Using the duality type estimates we can use the square function lower bounds to prove
+the inclusion of product type spaces.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+31
+7.5. Definition. Given a lattice of Zygmund rectangles DZ and a sequence of scalars
+B = (bI)I∈DZ we define
+∥B∥BMOprod := sup
+Ω
+�
+1
+|Ω|
+�
+I∈DZ
+I⊂Ω
+|bI|2
+� 1
+2
+.
+The inclusion of the little BMO space can be easily seen from the duality estimate
+(7.6)
+∥B∥BMOprod ∼ sup
+� �
+I∈DZ
+|aI||bI|:
+���
+� �
+I∈DZ
+|aI|2 1I
+|I|
+� 1
+2 ���
+L1 ≤ 1
+�
+.
+7.A. Paraproduct expansions. Here the correct expansions style is the Zygmund mar-
+tingale expansion similar to [14, Equation (5.22)]. This gives
+bf =
+�
+I∈DZ
+�
+∆I,Zb∆I,Zf + ∆I,Zb∆I1EI2,3f + ∆I1EI2,3b∆I,Zf
+(7.7)
++ ∆I,ZbEI1∆I2,3f + ∆I,ZbEI1EI2,3f + ∆I1EI2,3bEI1∆I2,3f
++ EI1∆I2,3b∆I,Zf + EI1∆I2,3b∆I1EI2,3f + EI1EI2,3b∆I,Zf
+�
+=:
+3
+�
+i,j=1
+ai,j(b, f),
+where, for example, a1,1 = �
+I∈DZ ∆I,Zb∆I,Zf and
+a1,2 =
+�
+I∈DZ
+∆I,Zb∆I1EI2,3f,
+i.e., interpret so that rows correspond to the first index i and columns correspond with
+the second index j.
+7.8. Lemma. If b ∈ bmoZ, then the paraproducts ai,j such that (i, j) ̸= (3, 3) are bounded. That
+is,
+∥ai,j(b, f)∥Lp ≲ ∥b∥bmoZ∥f∥Lp,
+1 < p < ∞.
+Proof. Case 1: product type i ̸= 3 ̸= j. We begin with the paraproducts where it would
+suffice to have a product BMO type assumption (but recall that little BMO is a subset).
+The symmetry Π = a1,1 is essentially trivial. By (7.6) we have
+|⟨Πf, g⟩| ≲
+���
+� �
+I∈DZ
+|⟨f, hI,Z⟩|2⟨|g|⟩2
+I
+1I
+|I|
+� 1
+2���
+L1
+≤
+���
+� �
+I∈DZ
+⟨|∆I,Zf|⟩2
+I1I
+� 1
+2 ���
+Lp∥MZg∥Lp′
+≲
+���
+� �
+I∈DZ
+MZ(∆I,Zf)2� 1
+2 ���
+Lp∥g∥Lp′
+≲ ∥SZf∥Lp∥g∥Lp′ ≲ ∥f∥Lp∥g∥Lp′ .
+
+32
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+The ‘twisted’ case Π = a1,2 (and the symmetrical a2,1) is trickier. Indeed, to decouple
+f and g we cannot blindly take maximal functions only in some parameters – this would
+break the Zygmund structure. In any case, we begin with the application of (7.6) to get
+|⟨Πf, g⟩| ≲
+���
+� �
+I∈DZ
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+��
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2 1I
+|I|
+� 1
+2 ���
+L1.
+The above is an L1 norm, while L2 would be nice. This is where A∞ extrapolation
+comes in. We fix ν ∈ A∞,Z, and move to estimate
+���
+� �
+I∈DZ
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+��
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2 1I
+|I|
+� 1
+2 ���
+L2(ν).
+We will soon show that
+���
+� �
+I∈DZ
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+��
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2 1I
+|I|
+� 1
+2 ���
+L2(ν)
+≲
+���MZf
+� �
+I1∈D1
+MZ(∆I1g)2�1/2���
+L2(ν).
+(7.9)
+The A∞ extrapolation, Theorem 7.10, then implies that this inequality holds also in Lp(ν),
+p ∈ (0, ∞), ν ∈ A∞,Z. We take p = 1 and ν ≡ 1 to get that
+|⟨Πf, g⟩| ≲
+���MZf
+� �
+I1∈D1
+MZ(∆I1g)2�1/2���
+L1
+≤ ∥MZf∥Lp
+���
+� �
+I1∈D1
+MZ(∆I1g)2�1/2���
+Lp′ ≲ ∥f∥Lp∥g∥Lp′ .
+It remains to prove (7.9). We write
+���
+� �
+I∈DZ
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+��
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2 1I
+|I|
+� 1
+2���
+2
+L2(ν)
+=
+�
+I1∈D1
+�
+I2×I3∈D2,3
+ℓ(I1)
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+����
+2���
+�
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2
+⟨ν⟩I.
+Fix some I1 ∈ D1. Let I2
+0 × I3
+0 ∈ D2,3
+ℓ(I1) and suppose ϕ1, ϕ2 and ϕ3 are locally inte-
+grable functions in R2. Then, there exists a sparse collection S = S(I2
+0 × I3
+0, ϕ1, ϕ2, ϕ3) ⊂
+D2,3
+ℓ(I1)(I2
+0 × I3
+0) so that
+�
+I2×I3∈D2,3
+ℓ(I1)
+I2×I3⊂I2
+0×I3
+0
+|⟨ϕ1, hI2×I3⟩|2|⟨ϕ2⟩I2×I3|2⟨ϕ3⟩I2×I3 ≲
+�
+Q∈S
+⟨|ϕ1|⟩2
+Q⟨|ϕ2|⟩2
+Q⟨|ϕ3|⟩Q|Q|.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+33
+We use this with the functions ϕ1 = ⟨f⟩I1, ϕ2 = ⟨g, hI1⟩ and ϕ3 = ⟨ν⟩I1 to have that for
+some sparse collection S = S(I1, I2
+0 × I3
+0, f, g, ν) ⊂ D2,3
+ℓ(I1) there holds that
+�
+I2×I3∈D2,3
+ℓ(I1)
+I2×I3⊂I2
+0×I3
+0
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+����
+2���
+�
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2
+⟨ν⟩I
+≲
+�
+Q∈S
+⟨|⟨f⟩I1|⟩2
+Q⟨|⟨g, hI1⟩|⟩2
+Q⟨ν⟩I1(Q)
+≤
+�
+Q∈S
+���
+M2,3
+ℓ(I1)⟨f⟩I1
+��
+M2,3
+ℓ(I1)⟨g, hI1⟩
+��⟨ν⟩I1
+Q
+�2
+⟨ν⟩I1(Q)
+≲
+ˆ
+R2
+�
+M2,3
+ℓ(I1)⟨f⟩I1
+�2�
+M2,3
+ℓ(I1)⟨g, hI1⟩
+�2⟨ν⟩I1,
+where in the last step we used the fact that ⟨ν⟩I1 ∈ A∞,ℓ(I1)(R2) and the Carleson embed-
+ding theorem.
+Since the last estimate holds uniformly for every I2
+0 × I3
+0 ∈ D2,3
+ℓ(I1), we get that
+�
+I1∈D1
+�
+I2×I3∈D2,3
+ℓ(I1)
+���
+�
+f, 1I1
+|I1| ⊗ hI2×I3
+����
+2���
+�
+g, hI1 ⊗
+1I2×I3
+|I2 × I3|
+����
+2
+⟨ν⟩I
+≲
+�
+I1∈D1
+ˆ
+R2
+�
+M2,3
+ℓ(I1)⟨f⟩I1
+�2�
+M2,3
+ℓ(I1)⟨g, hI1⟩
+�2⟨ν⟩I1
+≤
+�
+I1∈D1
+ˆ
+R3
+�
+M2,3
+ℓ(I1)⟨f⟩I1
+�2�
+M2,3
+ℓ(I1)⟨|∆I1g|⟩I1
+�21I1ν
+≤
+ˆ
+R2[MZf]2 �
+I1∈D1
+MZ(∆I1g)2ν.
+Thus, (7.9) is proved.
+Case 2: little BMO paraproducts (i = 3, j = 1, 2 or i = 1, 2, j = 3). Actually, now we only
+have “trivial” type cases with different twist. Symmetries a1,3 and a3,1 are similar as well
+as a2,3 and a3,2. Let us choose for example Π = a1,3 first. By Corollary 7.4 we have
+|⟨Π(b, f), g⟩| ≲
+���
+� �
+I1∈D1
+�
+�
+I2,3∈D2,3
+ℓ(I1)
+|⟨f, hI,Z⟩||⟨g, hI1hI1 ⊗ hI2,3⟩I| 1I2,3
+|I2,3|
+�2 1I1
+|I1|
+� 1
+2 ���
+L1.
+Now we again can use similar sparse method as above and for fixed I1 prove
+ˆ
+�
+I2,3∈Dℓ(I1)
+|⟨f, hI,Z⟩||⟨g, hI1hI1 ⊗ hI2,3⟩I| 1I2,3
+|I2,3|⟨ν⟩I1
+≲
+ˆ
+M2,3
+ℓ(I1)(⟨|∆I1f|⟩I1)M2,3
+ℓ(I1)⟨g⟩I11I1ν.
+
+34
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+The above estimate together with vector-valued version of Theorem 7.10 (proven in [3]
+for general Muckenhoupt basis) yields
+���
+� �
+I1∈D1
+�
+�
+I2,3∈Dℓ(I1)
+|⟨f, hI,Z⟩||⟨g, hI1hI1 ⊗ hI2,3⟩I| 1I2,3
+|I2,3|
+�2 1I1
+|I1|
+� 1
+2���
+L1
+≲
+���
+� �
+I1∈D1
+MZ(∆I1f)2 1I1
+|I1|
+� 1
+2MZg
+���
+L1
+≤
+���
+� �
+I1∈D1
+MZ(∆I1f)2�1/2���
+Lp∥MZg∥Lp′ ≲ ∥f∥Lp∥g∥Lp′ .
+Moving to the symmetry Π = a3,2 we first get
+|⟨Π(b, f), g⟩|
+=
+���
+�
+I∈DZ
+⟨⟨b⟩I1, hI2,3⟩
+�
+f, hI1 ⊗ 1I2,3
+|I2,3|
+�
+⟨g, hI,Z⟩
+���
+≲ ∥b∥bmoZ
+���
+�
+I1∈D1
+�
+�
+I2,3∈Dℓ(I1)
+|⟨f, hI1 ⊗ 1I2,3
+|I2,3|⟩|2|⟨g, hI,Z⟩|2 1I2,3
+|I2,3|
+� 1
+2 1I1
+|I1|
+���
+L1,
+where we use the other estimate in Corollary 7.4. Like above, we continue as follows
+���
+�
+I1∈D1
+�
+�
+I2,3∈Dℓ(I1)
+|⟨f, hI1 ⊗ 1I2,3
+|I2,3|⟩|2|⟨g, hI,Z⟩|2 1I2,3
+|I2,3|
+� 1
+2 1I1
+|I1|
+���
+L1
+≲
+���
+�
+I1∈D1
+M2,3
+ℓ(I1)⟨|∆I1f|⟩I1M2,3
+ℓ(I1)⟨|∆I1g|⟩I11I1
+���
+L1
+≤
+���
+� �
+I1∈D1
+MZ(∆I1f)2�1/2���
+Lp
+���
+� �
+I1∈D1
+MZ(∆I1g)2�1/2���
+Lp′
+≲ ∥f∥Lp∥g∥Lp′ .
+□
+In above proof we needed the A∞ extrapolation with Zygmund A∞ weights. In fact,
+we give a very simple proof of A∞ extrapolation [3] in general.
+7.10. Theorem. Let (f, g) be a pair of non-negative functions. Assume that there is some 0 <
+p0 < ∞ such that for all w ∈ A∞,Z there holds
+ˆ
+f p0w ≤ C([w]A∞,Z)
+ˆ
+gp0w,
+where C is an increasing function. Then for all 0 < p < ∞ and all w ∈ A∞,Z there holds
+ˆ
+f pw ≤ C([w]A∞,Z)
+ˆ
+gpw.
+Proof. We have for all 1 < r < ∞ and all w ∈ Ar,Z that
+ˆ
+(f p0/r)rw ≤ C([w]Ar,Z)
+ˆ
+(gp0/r)rw.
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+35
+Thus, by the classical extrapolation with Ap,Z weights we have
+(7.11)
+ˆ
+(f p0/r)sw ≤ C([w]As,Z)
+ˆ
+(gp0/r)sw
+for all 1 < s < ∞ and w ∈ As,Z.
+Finally, let 0 < p < ∞ and w ∈ A∞,Z. Then, there exists some 1 < s0 < ∞ such that
+w ∈ As0,Z. Choose some 1 < r < ∞ and s0 ≤ s < ∞ such that
+sp0/r = p.
+For example, we can take
+s = s0p
+p0
+�p0
+p + 1
+�
+= s0
+� p
+p0
++ 1
+�
+,
+r = s0
+�p0
+p + 1
+�
+.
+Since As0,Z ⊂ As,Z, we can use (7.11) with the exponents s and r to get the claim.
+□
+7.B. Zygmund shift commutators. Let k = (k1, k2), ki ∈ {0, 1, 2, . . .}, be fixed. A Zyg-
+mund shift Q = Qk of complexity k, see [14], has the form
+⟨Qkf, g⟩
+=
+�
+K∈D2−k1−k2+k3
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK⟨f, hI1 ⊗ HI2,3,J2,3⟩⟨g, HI1,J1 ⊗ hJ2,3⟩
+or
+⟨Qkf, g⟩
+=
+�
+K∈D2−k1−k2+k3
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK⟨f, hI1 ⊗ hI2,3⟩⟨g, HI1,J1 ⊗ HI2,3,J2,3⟩,
+where HI,J
+(1) is supported on I ∪ J and constant on children:
+HI,J =
+�
+L∈ch(I)∪ch(J)
+bL1L
+(2) is L2 normalized: |HI,J| ≤ |I|− 1
+2 , and
+(3) has zero average:
+´
+HI,J = 0.
+We will be focusing on the mixed type form since it is the most interesting one. Usually
+the other type is much easier and the method is easily recovered from this case.
+7.12. Proposition. Let Qk be a Zygmund shift of complexity k = (k1, k2, k3). Let 1 < p < ∞
+and b ∈ bmoZ . Then we have
+∥[b, Qk]f∥Lp ≲ max(k1, k2, k3)(|k| + 1)2∥b∥bmoZ∥f∥Lp.
+Proof. We consider the commutator [b, Qk]f : bQkf − Qk(bf) that in the dual form equals
+to
+�
+K∈D2−k1−k2+k3
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK
+�
+⟨bf, hI1 ⊗ HI2,3,J2,3⟩⟨g, HI1,J1 ⊗ hJ2,3⟩
+−⟨f, hI1 ⊗ HI2,3,J2,3⟩⟨bg, HI1,J1 ⊗ hJ2,3⟩
+�
+.
+
+36
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+Now, expanding both bf and bg with the expansion (7.7) we get the terms
+⟨Qk(ai,j(b, f)), g⟩
+and
+⟨Qkf, ai,j(b, g)⟩
+whenever (i, j) ̸= (3, 3). These terms are directly bounded separately, in particular, we
+have Qk : Lp → Lp and ai,j : Lp → Lp. Hence, we are left with bounding
+�
+K∈Dλ
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK
+� �
+L∈DZ
+⟨b⟩L⟨∆L,Zf, hI1 ⊗ HI2,3,J2,3⟩⟨g, HI1,J1 ⊗ hJ2,3⟩
+−
+�
+L∈DZ
+⟨b⟩L⟨f, hI1 ⊗ HI2,3,J2,3⟩⟨∆L,Zg, HI1,J1 ⊗ hJ2,3⟩
+�
+=
+�
+K∈Dλ
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK
+×
+�
+�
+L∈DZ
+ℓ(L1)=2−k1ℓ(K1)
+ℓ(K2)≤2k2ℓ(L2)≤2max(k2,k3)ℓ(K2)
+⟨b⟩L⟨∆L,Zf, hI1 ⊗ HI2,3,J2,3⟩⟨g, HI1,J1 ⊗ hJ2,3⟩
+−
+�
+Q∈DZ
+Q1⊂K1, ℓ(Q1)≥ℓ(I1)
+2−k1ℓ(K2)≤2k2ℓ(Q2)≤ℓ(K2)
+⟨b⟩Q⟨f, hI1 ⊗ HI2,3,J2,3⟩⟨∆Q,Zg, HI1,J1 ⊗ hJ2,3⟩
+�
+,
+where we have abbreviated 2−k1−k2+K3 by λ. Now, we write
+⟨f, hI1 ⊗ HI2,3,J2,3⟩ =
+�
+L∈DZ
+ℓ(L1)=2−k1ℓ(K1)
+ℓ(K2)≤2k2ℓ(L2)≤2max(k2,k3)ℓ(K2)
+⟨∆L,Zf, hI1 ⊗ HI2,3,J2,3⟩
+and
+⟨g, HI1,J1 ⊗ hJ2,3⟩ =
+�
+Q∈DZ
+Q1⊂K1, ℓ(Q1)≥ℓ(I1)
+2−k1ℓ(K2)≤2k2ℓ(Q2)≤ℓ(K2)
+⟨∆Q,Zg, HI1,J1 ⊗ hJ2,3⟩
+for the unexpanded terms. Thus, we end up with
+�
+K∈Dλ
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK
+�
+L∈DZ
+ℓ(L1)=2−k1ℓ(K1)
+ℓ(K2)≤2k2ℓ(L2)≤2max(k2,k3)ℓ(K2)
+�
+Q∈DZ
+Q1⊂K1, ℓ(Q1)≥ℓ(I1)
+2−k1ℓ(K2)≤2k2ℓ(Q2)≤ℓ(K2)
+×
+�
+(⟨b⟩L − ⟨b⟩Q)⟨∆L,Zf, hI1 ⊗ HI2,3,J2,3⟩⟨∆Q,Zg, HI1,J1 ⊗ hJ2,3⟩
+�
+.
+We write explicitly the complexity levels for Q and L. That is, in the above summations
+we have (L2)(l2) = (K2)(max(0,k3−k2)) for some l2 ∈ {0, . . . , max(k2, k3)}, (Q1)(q1) = K1,
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+37
+for some q1 ∈ {0, . . . , k1}, and (Q2)(q2) = K2 for some q2 ∈ {k2, . . . , k2 + k1}. We get
+�
+K∈Dλ
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK
+max(k2,k3)
+�
+l2=0
+�
+q1∈{0,...,k1}
+q2∈{k2,...,k2+k1}
+�
+L∈DZ
+ℓ(L1)=2−k1ℓ(K1)
+(L2)(l2)=(K2)(max(0,k3−k2))
+�
+Q∈DZ
+(Q1)(q1)=K1
+(Q2)(q2)=K2
+×
+�
+(⟨b⟩L − ⟨b⟩Q)⟨∆L,Zf, hI1 ⊗ HI2,3,J2,3⟩⟨∆Q,Zg, HI1,J1 ⊗ hJ2,3⟩
+�
+.
+Here we need to notice that R = R1 × R2 × R3 ⊃ K, L, Q, where
+R = K(k1,max(0,k3−k2),k1+max(k2−k3,0)) and R ∈ DZ.
+This is a common “Zygmund ancestor” for all of these rectangles.
+Let us expand in the difference ⟨b⟩L − ⟨b⟩Q in the following way
+⟨b⟩L = ⟨b⟩L − ⟨b⟩L(0,1,1)
++ ⟨b⟩L(0,1,1) − ⟨b⟩L(0,2,2)
+...
++ ⟨b⟩L(0,l2−1,l2−1) − ⟨b⟩L(0,l2,l2) + ⟨b⟩L(0,l2,l2)
+=
+l2−1
+�
+r2=0
+�
+⟨b⟩L(0,r2,r2) − ⟨b⟩L(0,r2+1,r2+1)
+�
++ ⟨b⟩L(0,l2,l2).
+Notice that since ℓ(L1)ℓ(L2) = ℓ(L3), we have ℓ(L1)ℓ((L2)(r2)) = ℓ((L3)(r2)), i.e. rectan-
+gles (L2)(r2) × (L3)(r2) ∈ Dℓ(L1) which is desirable since we want to use the characteriza-
+tion (2) in Lemma 7.2. We continue with the last term
+⟨b⟩L(0,l2,l2) = ⟨b⟩L(0,l2,l2) − ⟨b⟩L(1,l2,1+l2)
++ ⟨b⟩L(1,l2,1+l2) − ⟨b⟩L(2,l2,2+l2)
+...
+⟨b⟩L(k1−1,l2,k1−1+l2) − ⟨b⟩L(k1,l2,k1+l2) + ⟨b⟩G
+=
+k1−1
+�
+r1=0
+�
+⟨b⟩L(r1,l2,r1+l2) − ⟨b⟩L(r1+1,l2,r1+1+l2)
+�
++ ⟨b⟩R.
+Recall that (L2)(l2) = (K2)(max(0,k3−k2)) =: R2 and observe that since ℓ((L3)(k1+l2)) =
+ℓ((L2)(l2))ℓ((L1)(k1)) = ℓ(R2)ℓ(K1) we get (L3)(k1+l2) = R3. Thus, we end up with a sum
+of terms of the forms
+⟨b⟩L(0,r2,r2) − ⟨b⟩L(0,r2+1,r2+1)
+and
+⟨b⟩L(r1,l2,r1+l2) − ⟨b⟩L(r1+1,l2,r1+1+l2),
+(7.13)
+and we have for fixed r1 and r2
+|(7.13)| ≲ ∥b∥bmoZ
+by Lemma 7.2.
+
+38
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+By the same argument as above we get
+⟨b⟩Q =
+max(0,k3−k2)+q2−1
+�
+ρ2=0
+⟨b⟩Q(0,ρ2,ρ2) − ⟨b⟩Q(0,ρ2+1,ρ2+1)
++
+q1
+�
+ρ1=0
+⟨b⟩Q(ρ1,�q2,ρ1+�q2) − ⟨b⟩Q(ρ1+1,�q2,ρ1+1+�q2)
++ ⟨b⟩R,
+where �q2 = max(0, k3 − k2) + q2,
+(Q2)(�q2) = (K2)(max(0,k3−k2))
+and
+(Q3)(q1+�q2) = (K3)(k1+max(k2−k3,0)).
+Notice that the last term corresponds to the last term in the previous expansion, and
+hence, their difference equals to zero. Again, here we have
+|⟨b⟩Q(0,ρ2,ρ2) − ⟨b⟩Q(0,ρ2+1,ρ2+1) + ⟨b⟩Q(ρ1,�q2,ρ1+�q2) − ⟨b⟩Q(ρ1+1,�q2,ρ1+1+�q2)| ≲ ∥b∥bmoZ
+for fixed ρ1 and ρ2.
+Now, we can split the commutator into the two terms
+Wb
+K,kf = 1K
+�
+L∈DZ
+ℓ(L1)=2−k1ℓ(K1)
+ℓ(K2)≤2k2ℓ(L2)≤2max(k2,k3)ℓ(K2)
+bL,K∆L,Zf,
+where
+|bL,K| ≲ max(k1, k2, k3)∥b∥bmoZ,
+and
+Vb
+K,kg =
+�
+Q∈DZ
+Q1⊂K1, ℓ(Q1)≥ℓ(I1)
+2−k1ℓ(K2)≤2k2ℓ(Q2)≤ℓ(K2)
+bQ,K∆Q,Zg,
+where
+|bQ,K| ≲ max(k1, k2, k3)∥b∥bmoZ.
+Thus, the last term of the commutator is the sum of
+�
+K∈Dλ
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK⟨Wb
+K,kf, hI1 ⊗ HI2,3,J2,3⟩⟨VK,kg, HI1,J1 ⊗ hJ2,3⟩
+and
+�
+K∈Dλ
+�
+I,J∈DZ
+I(k)=K=J(k)
+aIJK⟨WK,kf, hI1 ⊗ HI2,3,J2,3⟩⟨Vb
+K,kg, HI1,J1 ⊗ hJ2,3⟩.
+The boundedness follows via standard methods (adapt proofs of [14, Theorem 6.2 and
+Lemma 5.20].)
+□
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+39
+APPENDIX A. BILINEAR FEFFERMAN-PIPHER MULTIPLIERS
+In this section we consider bilinear variants of multipliers studied by Fefferman-Pipher [6].
+These considerations motivate the kernel estimates in Section 3. After the presented cal-
+culations, the reader can easily check how everything fits with Section 3. In fact, we will
+see that the bilinear Fefferman-Pipher multipliers produce kernels which satisfy the the
+kernel estimates in Section 3 with
+θ = 2,
+α1 = 1,
+α2,3 = 1,
+and an extra logarithm factor. In the partial kernel estimates �θ = 1 and there is also a
+harmless logarithm factor. We leave further analysis of these multipliers for future work.
+We consider the following multi-parameter dilation on R6 – define
+ρs,t(x, y) = (sx1, tx2, stx3, sy1, ty2, sty3),
+s, t > 0,
+and set
+A1 := {(ξ, η) ∈ R6 : 1
+2 < |(ξ1, η1)| ≤ 1, 1
+2 < |(ξ2, ξ3, η2, η3)| ≤ 1}.
+In this section we consider the parameter groups {1} and {2, 3} only. The grouping
+{{2}, {1, 3}} is similar, for example, we would set
+A2 := {(ξ, η) ∈ R6 : 1
+2 < |(ξ2, η2)| ≤ 1, 1
+2 < |(ξ1, ξ3, η1, η3)| ≤ 1}.
+For Schwartz functions f1, f2 we define the bilinear multiplier operator
+Tm,1(f1, f2)(x) =
+ˆ
+R3
+ˆ
+R3 m(ξ, η) �f1(ξ) �f2(η)e2πix·(ξ+η) dξ dη,
+where the symbol m ∈ CN is assumed to satisfy
+∥m∥M1
+Z :=
+sup
+|α|∞≤N
+|β|∞≤N
+sup
+s,t>0
+sup
+(ξ,η)∈A1 |∂α
+ξ ∂β
+η (m ◦ ρs,t)(ξ, η)| < ∞.
+Thus, if (ξ, η) ∈ A1, then by definition
+|(∂α
+ξ ∂β
+η m)(sξ1, tξ2, stξ3, sη1, tη2, stη3)| ≤ ∥m∥M1
+Zs−α1−β1t−α2−β2(st)−α3−β3
+(A.1)
+= ∥m∥M1
+Zs−(α1+β1)+(α2+β2)(st)−(α2+β2)−(α3+β3).
+Now, for (ζ1, σ1) ̸= 0 and (ζ2, ζ3, σ2, σ3) ̸= 0 denote
+s = |(ζ1, σ1)|,
+st = |(sζ2, ζ3, sσ2, σ3)|,
+(ξ1, ξ2, ξ3) =
+�ζ1
+s , ζ2
+t , ζ3
+st
+�
+,
+(η1, η2, η3) =
+�σ1
+s , σ2
+t , σ3
+st
+�
+.
+Thus, (ξ, η) ∈ A1 and
+|∂α
+ζ ∂β
+σm(ζ, σ)| ≲ ∥m∥M1
+Z(|ζ1| + |σ1|)−(α1+β1)+(α2+β2)
+(A.2)
+×
+�
+|((|ζ1| + |σ1|)ζ2, ζ3)| + |((|ζ1| + |σ1|)σ2, σ3)|
+�−(α2+β2)−(α3+β3).
+We write, with two standard partition of unity φ1 on R2 \{0} and φ2,3 on R4 \{0}, that
+1 =
+�
+j,k∈Z
+φ1(2−jξ1, 2−jη1)φ2,3(2−kξ2, 2−j−kξ3, 2−kη2, 2−j−kη3).
+
+40
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+Via this identity we obtain
+m =
+�
+j,k
+(φ1 ⊗ φ2,3 ◦ ρ2−j,2−k) · m
+=
+�
+j,k
+(φ1 ⊗ φ2,3 · (m ◦ ρ2j,2k)) ◦ ρ2−j,2−k =: mj,k.
+Since φ1 and φ2,3 are supported in ¯B(0, 2) \ B(0, 1
+2) in R2 and R4, respectively, we know
+that
+spt mj,k ⊂
+ρ2j,2k
+�
+(ξ, η) : (ξ1, η1) ∈ ¯BR2(0, 2) \ BR2(0, 1
+2), (ξ2,3, η2,3) ∈ ¯BR4(0, 2) \ BR4(0, 1
+2)
+�
+.
+Using this we get
+∥∂α∂βmj,k∥L∞ ≲ 2−(j,k,j+k)·(α+β)
+and
+∥∂α∂βmj,k∥L1 ≲ 2(j,k,j+k)·(2−(α+β)),
+where 2 = (2, 2, 2).
+Let Kj,k(y, z) = ˇmj,k and K(y, z) = �
+j,k Kj,k(y, z) – then K(x − y, x − z) is the corre-
+sponding kernel. Using similar analysis as in [14] we have
+∥yαz ˜α∂β
+y ∂γ
+z Kj,k∥L∞ ≲ ∥∂α
+ξ ∂ ˜α
+η (ξβηγmj,k)∥L1
+≤
+�
+l≤α
+˜l≤˜α
+�α
+l
+��˜α
+˜l
+�
+∥∂l(ξβ)∂
+˜l(ηγ) · ∂α−l∂ ˜α−˜lmj,k)∥L1
+≲ 2(j,k,j+k)·(2+(β+γ)−(α+˜α))
+for multi-indices α, ˜α, β, γ. Hence, we get
+|yβ+1zγ+1∂β
+y ∂γ
+z Kj,k(y, z)| ≲ 2(j,k,j+k)·(2+(β+γ)−(α+˜α))|yβ+1−α| · |zγ+1−˜α|.
+Taking αi, ˜αi ∈ {0, N} we obtain
+|yβ+1zγ+1∂β
+y ∂γ
+z K(y, z)|
+≲
+�
+j
+min{(2j|y1|)β1+1, (2j|y1|)β1+1−N} min{(2j|z1|)γ1+1, (2j|z1|)γ1+1−N}
+×
+�
+k
+min{(2k|y2|)β2+1, (2k|y2|)β2+1−N} min{(2k|z2|)γ2+1, (2k|z2|)γ2+1−N}
+× min{(2j+k|y3|)β3+1, (2j+k|y3|)β3+1−N} min{(2j+k|z3|)γ3+1, (2j+k|z3|)γ3+1−N}.
+We can estimate the inner sum either by
+�
+k : 2k<1/(|y2|+|z2|)
+(2k|y2|)β2+1(2k|z2|)γ2+1(2j+k|y3|)β3+1(2j+k|z3|)γ3+1
++
+�
+k : 2k≥1/(|y2|+|z2|)≥1/(2|y2|)
+(2k|y2|)β2+1−N(2k|z2|)γ2+1(2j+k|y3|)β3+1(2j+k|z3|)γ3+1
++
+�
+k : 2k≥1/(|y2|+|z2|)>1/(2|z2|)
+(2k|y2|)β2+1(2k|z2|)γ2+1−N(2j+k|y3|)β3+1(2j+k|z3|)γ3+1
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+41
+≲
+|y2|β2+1
+(|y2| + |z2|)β2+1 ·
+|z2|γ2+1
+(|y2| + |z2|)γ2+1 ·
+(2j|y3|)β3+1
+(|y2| + |z2|)β3+1 ·
+(2j|z3|)γ3+1
+(|y2| + |z2|)γ3+1 =: I1
+or by
+�
+k : 2k<2−j/(|y3|+|z3|)
+(2k|y2|)β2+1(2k|z2|)γ2+1(2j+k|y3|)β3+1(2j+k|z3|)γ3+1
++
+�
+k : 2k≥2−j/(|y3|+|z3|)≥2−j/(2|y3|)
+(2k|y2|)β2+1(2k|z2|)γ2+1(2j+k|y3|)��3+1−N(2j+k|z3|)γ3+1
++
+�
+k : 2k≥2−j/(|y3|+|z3|)>2−j/(2|z3|)
+(2k|y2|)β2+1(2k|z2|)γ2+1(2j+k|y3|)β3+1(2j+k|z3|)γ3+1−N
+≲
+|y2|β2+1
+[2j(|y3| + |z3|)]β2+1 ·
+|z2|γ2+1
+[2j(|y3| + |z3|)]γ2+1 ·
+|y3|β3+1
+(|y3| + |z3|)β3+1 ·
+|z3|γ3+1
+(|y3| + |z3|)γ3+1 =: I2,
+in both cases provided that β2 + β3 + γ2 + γ3 < N − 4.
+The outer sum can then be estimated either by
+�
+j : 2j<1/(|y1|+|z1|)
+(2j|y1|)β1+1(2j|z1|)γ1+1I1
++
+�
+j : 2j≥1/(|y1|+|z1|)≥1/(2|y1|)
+(2j|y1|)β1+1−N(2j|z1|)γ1+1I1
++
+�
+j : 2j≥1/(|y1|+|z1|)>1/(2|z1|)
+(2j|y1|)β1+1(2j|z1|)γ1+1−NI1
+≲
+|y1|β1+1|z1|γ1+1
+(|y1| + |z1|)β1+γ1+2
+|y2|β2+1|z2|γ2+1
+(|y2| + |z2|)β2+γ2+2
+|y3|β3+1|z3|γ3+1
+[(|y1| + |z1|)(|y2| + |z2|)]β3+γ3+2
+or, if (|y1| + |z1|)(|y2| + |z2|) ≤ |y3| + |z3|, by
+�
+j : 2j<(|y2|+|z2|)/(|y3|+|z3|)
+(2j|y1|)β1+1(2j|z1|)γ1+1I1
++
+�
+j : |y2|+|z2|
+|y3|+|z3| ≤2j≤
+1
+|y1|+|z1|
+(2j|y1|)β1+1(2j|z1|)γ1+1I2
++
+�
+j : 2j>1/(|y1|+|z1|)>1/(2|z1|)
+(2j|y1|)β1+1(2j|z1|)γ1+1−NI2
++
+�
+j : 2j>1/(|y1|+|z1|)>1/(2|y1|)
+(2j|y1|)β1+1−N(2j|z1|)γ1+1I2 =: I + II + III + IV.
+It is straightforward that
+I ∼
+|y1|β1+1|z1|γ1+1
+(|y1| + |z1|)β1+γ1+2
+|y2|β2+1|z2|γ2+1
+(|y2| + |z2|)β2+γ2+2
+|y3|β3+1|z3|γ3+1
+(|y3| + |z3|)β3+γ3+2
+×
+�(|y1| + |z1|)(|y2| + |z2|)
+|y3| + |z3|
+�β1+γ1+2
+;
+III ∼ IV ∼
+|y1|β1+1|z1|γ1+1
+(|y1| + |z1|)β1+γ1+2
+|y2|β2+1|z2|γ2+1
+(|y2| + |z2|)β2+γ2+2
+|y3|β3+1|z3|γ3+1
+(|y3| + |z3|)β3+γ3+2
+
+42
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+×
+�(|y1| + |z1|)(|y2| + |z2|)
+|y3| + |z3|
+�β2+γ2+2
+.
+Lastly, we have
+II ∼
+|y1|β1+1|z1|γ1+1
+(|y1| + |z1|)β1+γ1+2
+|y2|β2+1|z2|γ2+1
+(|y2| + |z2|)β2+γ2+2
+|y3|β3+1|z3|γ3+1
+(|y3| + |z3|)β3+γ3+2
+×
+�(|y1| + |z1|)(|y2| + |z2|)
+|y3| + |z3|
+�min{β1+γ1,β2+γ2}+2
+Lβ1,β2,γ1,γ2(y, z),
+where
+Lβ1,β2,γ1,γ2(y, z) := 1 + log+
+�
+|y3| + |z3|
+(|y1| + |z1|)(|y2| + |z2|)
+�
+when β1 + γ1 = β2 + γ2 and Lβ1,β2,γ1,γ2(y, z) = 1 otherwise. In conclusion, we get
+|∂β
+y ∂γ
+z K(y, z)| ≲
+1
+[(|y1| + |z1|)(|y2| + |z2|) + |y3| + |z3|]β3+γ3+4
+×
+1
+(|y1| + |z1|)β1+γ1(|y2| + |z2|)β2+γ2
+× min
+�
+1,
+�(|y1| + |z1|)(|y2| + |z2|)
+|y3| + |z3|
+�min{β1+γ1,β2+γ2}�
+Lβ1,β2,γ1,γ2(y, z).
+1.A. Partial kernel estimates. Let m ∈ M1
+Z. We define truncations of m by setting
+mJ :=
+�
+|j|≤J1,|k|≤J2
+mj,k,
+J = (J1, J2) ∈ N2.
+A.3. Lemma. Suppose that m ∈ M1
+Z. Let mJ be defined as above and let KJ = ˇmJ. Then for
+(y2, z2) ̸= 0 ̸= (y3, z3) we have the estimate
+���
+˚
+I1×I1×I1 ∂β2
+y2 ∂β3
+y3 ∂γ2
+z2 ∂γ3
+z3 KJ(x1 − y1, y2, y3, x1 − z1, z2, z3) dy1 dz1 dx1
+���
+≲
+1
+(|y2| + |z2|)β2+γ2 ·
+1
+(|y3| + |z3|)β3+γ3 |I1|(|I1|(|y2| + |z2|)
+|y3| + |z3|
++
+|y3| + |z3|
+|I1|(|y2| + |z2|))−1
+×
+1
+�3
+i=2(|yi| + |zi|)2 ·
+�
+1 + log+
+|y3| + |z3|
+|I1|(|y2| + |z2|)
+�
+,
+where I1 is an interval and β2 + β3 + γ2 + γ3 ≤ 1.
+Proof. Since mJ(0, ξ2, ξ3, 0, η2, η3) = 0, using the Fourier transform we know that
+(A.4)
+¨
+R2 ∂β2
+y2 ∂β3
+y3 ∂γ2
+z2 ∂γ3
+z3 KJ(y1, y2, y3, z1, z2, z3) dy1 dz1 = 0.
+Suppose first that |I1|(|y2| + |z2|) ≥ |y3| + |z3| – by (A.4) we may equivalently estimate
+the integral over I1 × (R2 \ (I1 × I1)) instead of I1 × I1 × I1. By the kernel estimates we
+have ���
+˚
+I1×(R2\(I1×I1))
+∂β2
+y2 ∂β3
+y3 ∂γ2
+z2 ∂γ3
+z3 KJ(x1 − y1, y2, y3, x1 − z1, z2, z3) dy1 dz1 dx1
+���
+≲
+ˆ
+I1
+¨
+R2\(I1×I1)
+1
+(|y2| + |z2|)β2+γ2
+
+ZYGMUND DILATIONS: BILINEAR ANALYSIS AND COMMUTATOR ESTIMATES
+43
+×
+1 + log+
+|y3|+|z3|
+(|x1−y1|+|x1−z1|)(|y2|+|z2|)
+[(|x1 − y1| + |x1 − z1|)(|y2| + |z2|) + |y3| + |z3|]β3+γ3+4 dy1 dz1 dx1.
+Note that we have either y1 ∈ R\I1 or z1 ∈ R\I1, and we may without loss of generality
+assume y1 ∈ R \ I1. Then the integral is dominated by
+ˆ
+I1
+¨
+(R\I1)×R
+1
+(|y2| + |z2|)β2+γ2
+×
+1 + log+
+|y3|+|z3|
+|x1−y1|(|y2|+|z2|)
+[(|x1 − y1| + |x1 − z1|)(|y2| + |z2|) + |y3| + |z3|]β3+γ3+4 dy1 dz1 dx1
+≲
+1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+ˆ
+I1
+ˆ
+R\I1
+1 + log+
+|y3|+|z3|
+|x1−y1|(|y2|+|z2|)
+�
+|x1 − y1| + |y3|+|z3|
+|y2|+|z2|
+�β3+γ3+3 dy1 dx1.
+Let t := |y3|+|z3|
+|y2|+|z2|. By a change of variables we reduce to
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+¨
+t−1I1×(R\t−1I1)
+1 + log+
+1
+|x1−y1|
+�
+|x1 − y1| + 1
+�β3+γ3+3 dy1 dx1
+≲
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+ˆ
+t−1I1
+1
+�
+d(x1, ∂(t−1I1)) + 1
+�β3+γ3+2 dx1
+≲
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+=
+1
+(|y2| + |z2|)β2+γ2+3
+1
+(|y3| + |z3|)β3+γ3+1
+∼
+1
+(|y2| + |z2|)β2+γ2 ·
+1
+(|y3| + |z3|)β3+γ3 |I1|(|I1|(|y2| + |z2|)
+|y3| + |z3|
++
+|y3| + |z3|
+|I1|(|y2| + |z2|))−1
+×
+1
+�3
+i=2(|yi| + |zi|)2 .
+Assume then that |I1|(|y2| + |z2|) < |y3| + |z3|. This time we integrate over I1 × I1 × I1.
+Proceeding as above we reduce to the integral
+˚
+t−1I1×t−1I1×t−1I1
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+1 + log+
+1
+(|x1−y1|+|x1−z1|)
+[(|x1 − y1| + |x1 − z1|) + 1]β3+γ3+4 dy1 dz1 dx1
+≤
+¨
+t−1I1×t−1I1
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+1 + log+
+1
+|x1−y1|
+(|x1 − y1| + 1)β3+γ3+3 dy1 dx1
+∼
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4
+¨
+t−1I1×t−1I1
+�
+1 + log+
+1
+|x1 − y1|
+�
+dy1 dx1
+≲
+t−β3−γ3−1
+(|y2| + |z2|)β2+γ2+β3+γ3+4 (t−1|I1|)2(1 + log+(t|I1|−1))
+
+44
+EMIL AIRTA, KANGWEI LI, AND HENRI MARTIKAINEN
+=
+1
+(|y2| + |z2|)β2+γ2 ·
+1
+(|y3| + |z3|)β3+γ3 |I1|(|I1|(|y2| + |z2|)
+|y3| + |z3|
++
+|y3| + |z3|
+|I1|(|y2| + |z2|))−1
+×
+1
+�3
+i=2(|yi| + |zi|)2 ·
+�
+1 + log+
+|y3| + |z3|
+|I1|(|y2| + |z2|)
+�
+.
+Thus, we are done.
+□
+With (A.2) at hand, similarly as in the linear case we can derive the following.
+A.5. Lemma. Let m ∈ M1
+Z and denote by Tm the corresponding Fourier multiplier operator.
+Let f1, g1 ∈ L4(R), f2,3, g2,3 ∈ L4(R2) and h1 ∈ L2(R), h2,3 ∈ L2(R2). Then
+⟨Tm(f1 ⊗ f2,3, g1 ⊗ g2,3), h1 ⊗ h2,3⟩ = ⟨Tmf2,3,g2,3,h2,3(f1, g1), h1⟩,
+where mf2,3,g2,3,h2,3 is a standard bilinear Coifman-Meyer multiplier in R satisfying the estimates
+|( d/ dξ1)α( d/ dη1)βmf2,3,g2,3,h2,3(ξ1, η1)|
+≲ ∥m∥M1
+Z∥f2,3∥L4∥g2,3∥L4∥h2,3∥L2(|ξ1| + |η1|)−α−β.
+Thus, Tmf2,3,g2,3,h2,3 is a convolution form bilinear Calderón-Zygmund operator. In particular,
+there exists a standard bilinear Calderón-Zygmund kernel Km,f2,3,g2,3,h2,3 such that
+∥Km,f2,3,g2,3,h2,3∥CZ1(R2) ≲ ∥f2,3∥L4∥g2,3∥L4∥h2,3∥L2.
+Moreover, if spt f1 ∩ spt g1 ∩ spt h1 = ∅, then
+⟨Tm(f1 ⊗ f2,3, g1 ⊗ g2,3), h1 ⊗ h2,3⟩
+=
+˚
+Km,f2,3,g2,3,h2,3(x1, y1, z1)f1(y1)g1(z1)h1(x1) dy1 dz1 dx1.
+REFERENCES
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