# Unconventional superconducting pairing in a B20 Kramers Weyl semimetal

Sougata Mardanya<EMAIL_ADDRESS>Department of Physics and
Astrophysics, Howard University, Washington, USA  Mehdi Kargarian
<EMAIL_ADDRESS>Department of Physics, Sharif University of Technology,
Tehran, Iran  Rahul Verma  Department of Condensed Matter Physics and
Materials Science, Tata Institute of Fundamental Research, Mumbai 400005,
India  Tay-Rong Chang  Department of Physics, National Cheng Kung University,
Tainan 701, Taiwan  Center for Quantum Frontiers of Research and Technology,
Tainan 701, Taiwan  Physics Division, National Center for Theoretical
Sciences, National Taiwan University, Taipei, Taiwan  Sugata Chowdhury
Department of Physics and Astrophysics, Howard University, Washington, USA
Hsin Lin  Institute of Physics, Academia Sinica, Taipei 11529, Taiwan  Arun
Bansil  Department of Physics, Northeastern University, Boston, Massachusetts
02115, USA  Amit Agarwal  Department of Physics, Indian Institute of
Technology, Kanpur 208016, India  Bahadur Singh<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India

###### Abstract

Topological superconductors present an ideal platform for exploring nontrivial
superconductivity and realizing Majorana boundary modes in materials. However,
finding a single-phase topological material with nontrivial superconducting
states is a challenge. Here, we predict nontrivial superconductivity in the
pristine chiral metal RhGe with a transition temperature of 5.8 K. Chiral
symmetries in RhGe enforce multifold Weyl fermions at high-symmetry momentum
points and spin-polarized Fermi arc states that span the whole surface
Brillouin zone. These bulk and surface chiral states support multiple type-II
van Hove singularities that enhance superconductivity in RhGe. Our detailed
analysis of superconducting pairing symmetries involving Chiral Fermi pockets
in RhGe indicates the presence of nontrivial superconducting pairing. Our
study establishes RhGe as a promising candidate material for hosting mixed-
parity pairing and topological superconductivity.

The quest for Majorana zero modes (MZMs) which are crucial for the development
of fault-tolerant quantum computation devices, has spurred significant
interest in the search for topological superconductors [1, 2, 3]. Recent
efforts have predominantly focused on characterizing MZMs in theoretical
models and on experiments with complex heterostructures [4, 5, 6, 7].
Unfortunately, the observation of isolated MZMs is still inconclusive. In
their simplest form, the MZMs have been predicted to exist at the boundary of
a two-dimensional chiral $p-$wave superconductor [8, 9, 10]. However, the
scarcity of such superconductors necessitates the exploration of alternative
avenues for realizing MZMs.

A significant breakthrough is the prediction of MZMs at the vortex core of the
gapped surface states of three-dimensional (3D) topological insulators under
the proximity effect of an $s$-wave superconductor [11, 12]. This has ignited
efforts to look for superconductivity in topological insulators and semimetals
employing chemical doping, external pressure, and reduced dimensionality [13,
14, 15, 16, 17, 18, 19, 20, 21, 22]. Noteworthy examples are the iron-based
superconductors with Dirac-type inverted band crossings, which show the
signature of topological superconductivity with MZMs at the vortex core [23,
24, 25]. Alternately, the momentum-dependent odd-parity pairing can originate
from the large spin-orbit coupling (SOC) in non-centrosymmetric
superconductors [26, 27, 28, 29, 30, 31, 32]. However, most superconductors
reported to date are inversion symmetric. The Pauli exclusion principle and
parity conservation dictate that a superconductor with even parity must be a
spin singlet, while an odd parity superconductor must be spin-triplet [33,
34]. A material that lacks inversion symmetry and has parity-breaking SOC can
thus support an admixture of spin-singlet and spin-triplet paring [31, 27, 28,
29, 30, 32].

Motivated by this, we show that the CoSi family of materials are ideal
candidates for hosting topological superconductivity. These materials form
chiral crystal structures and support unconventional Kramers-Weyl fermions
[35, 36, 37, 38, 39]. They host the longest possible Fermi-arc surface states
spanning the whole surface Brillouin zone (BZ) and feature intriguing
optoelectronic and spintronic properties, including helicity-dependent
photocurrent [40], a large circular photogalvanic effect [41], a spin Nernst
effect [42], and exciting plasmons [43]. Here, we demonstrate that RhGe
exhibits an electron-phonon coupling (EPC) mediated superconductivity driven
by multiple type-II van Hove singularities (VHSs) associated with its chiral
bands. The superconducting state, nontrivial electronic topology, and chiral
crystal symmetries offer all the crucial prerequisites for hosting mixed-
parity topological superconductivity in RhGe [44, 45, 46, 47]. By
characterizing possible pairing symmetries in Fermi pockets around Kramer Weyl
points ($\Gamma$ and $R$ points), we demonstrate a time-reversal symmetry
broken topological superconducting state in RhGe.

Beginning with structural chirality and multifold fermions, RhGe forms a
B20-type chiral cubic lattice with space group $\mathrm{P2_{1}3}$ (Fig. 1(a))
[48, 49]. The B20 structure emerges due to a pair sublattice distortion along
the $[111]$ direction (chiral axis), which dimerizes the bonds and breaks
(roto)-inversion symmetries [Figs. 1(a)-(b)]. It preserves a three-fold
rotation $\mathrm{\\{C_{3,111}|000\\}}$ symmetry along $[111]$ direction and
three $\frac{2\pi}{3}$ screw rotation symmetries with a non-primitive
translation ($\mathrm{\\{S_{2,100}|\frac{1}{2}\frac{1}{2}0\\}}$,
$\mathrm{\\{S_{2,010}|0\frac{1}{2}\frac{1}{2}\\}}$, and
$\mathrm{\\{S_{2,001}|\frac{1}{2}0\frac{1}{2}\\}}$). The fully relaxed lattice
parameter of RhGe is $a=4.79~{}\mathrm{\AA}$ and internal parameters are
$u_{Rh}=0.134$ and $u_{Ge}=0.838$ (both Rh and Ge occupy the Wyckoff position
$4a\rightarrow\\{u_{x},u_{x},u_{x}\\}$, $x$ is Rh or Ge), see Supplementary
Materials (SMs) for details.

Figure 1: (a)-(b) Chiral crystal structure of RhGe with a pair distortion
along [111] chiral axis. (c) The bulk Brillouin zone (BZ) and the projected
(001) surface BZ. The high symmetry $k$ points, chiral charges, and the Fermi-
arc surface states are highlighted. (d) Bulk band structure of RhGe with spin-
orbit coupling. The Rh-$d$ and Ge-($s,p$) orbital contributions and band
representations are shown. (e) The (001)-surface spectral function along the
high symmetry directions in the surface BZ. Sharp black lines show the chiral
surface states. (f) Calculated band countors at the Fermi level. The Fermi-arc
surface states connect the +4 chiral charge Weyl node at the BZ center
$\overline{\Gamma}$ with the -4 chiral charge node at the BZ corner
$\overline{M}$. The top and bottom panels depict the opposite connectivity of
Fermi-arc states at the top (Rh termination) and bottom (Ge termination)
surfaces.

The bulk band structure of RhGe is shown in Fig. 1(d). The presence of time-
reversal and non-symmorphic symmetries enforces multiple band degeneracies to
generate a unique semi-metallic phase with unconventional multifold fermions.
In the absence of SOC, the band structure realizes a three-fold and a four-
fold degenerate band-crossing pinned at the high-symmetry $\Gamma$ and $R$
points with chiral charges of +2 and -2, respectively. The inclusion of the
SOC splits the three-fold node at $\Gamma$ into a Kramers Weyl node at -0.48
eV and a four-fold degenerate unconventional RSW Weyl node at -0.37 eV. At the
$R$ point, a six-fold degenerate double spin-1 chiral node is formed at -0.58
eV. The chiral charges of the nodes at the $\Gamma$ and $R$ points are $+4$
and $-4$, respectively. The surface state spectrum given in Fig. 1(e) shows
four topological surface states connecting the projection of the bulk chiral
nodes at the $\overline{\Gamma}$ and $\overline{R}$ points. At the Fermi
energy, these states form the Fermi arcs spanning the entire BZ. Unlike the
surface states of generic-point Weyl semimetals, the surface states in RhGe
originate from the symmetry-enforced bulk Weyl nodes, making them robust
against local perturbations and disorder. The chiral nature of these surface
states is depicted in the electronic structure of the top and bottom surface
terminations, where the Fermi-arc-connectivity is seen to switch between the
pockets at the center and corners of the surface BZ [Fig. 1(f)].

Figure 2: (a) Spin-polarized surface states of RhGe on the (001) surface along
$\overline{X}-\overline{\Gamma}-\overline{M}-\overline{X}$. The dashed blue
lines c1 (17 meV), c2 (30 meV), c3 (50 meV), and c4 (160 meV) below the Fermi
level show constant energy cuts in (b)-(e). The chiral fermi-arc states
exchange connectivity from c1 to c4. Red arrows in c2 and c4 locate the
generic $k$-points type II VHSs that appear as a crossing between chiral
surface states.

In Fig. 2, we demonstrate the evolution of surface Fermi arcs as a function of
band energy and reveal the presence of type-II VHSs as seen experimentally in
CoSi and RhSi [50]. Note that the position of the VHS at generic $k$ point
dictates the nature of the VHS as type II. Figure 2(a) illustrates the
calculated spin-polarized surface states along
$\overline{X}-\overline{\Gamma}-\overline{M}-\overline{X}$. A large SOC-driven
spin-splitting in the chiral surface states is evident with high spin
polarization at the Fermi energy. The evolution of the associated Fermi arcs
is shown in Figs. 2(b)-(e) through successive energy cuts, where the VHSs
appear as crossing points of different pockets. In the constant-energy cut c1
($E=17$ meV), one interband and one intraband crossings appear around the
$\overline{X}$ and $\overline{Y}$ points, respectively. As we lower energy to
30 meV (cut c2), the two helicoid arc states move toward each other and touch
at the generic momentum points $(\pm 0.49,\pm 0.19)~{}\mathrm{\AA}^{-1}$ to
form type-II VHSs. Further lowering the energy to 50 meV (cut c3) captures the
intermediate stage of the chirality exchange between the helicoid states,
where the inner state along $\overline{\Gamma}-\overline{Y}$ cross at $(0,\pm
0.52)~{}\mathrm{\AA}^{-1}$. Finally, at an energy 160 meV (cut c4), another
intraband crossing appears at symmetry equivalent generic momentum point $(\pm
0.26,\mp 0.02)~{}\mathrm{\AA}^{-1}$. These multiple type-II VHSs in the
surface state spectrum could induce surface electronic instability and lead to
unconventional superconductivity in RhGe.

Figure 3: (a) Bulk band structure of RhGe connecting unconventional Weyl
fermions at the $\Gamma$ and $R$ points (left panel) and electronic density of
states (right panel). Location of bulk VHSs on the $\Gamma-R$ line is marked.
(b) The constant energy surface at $k_{z}=0.62~{}\mathrm{\AA}$. Locations of
VHS in the bulk BZ are marked with red circles. (c) Three-dimensional
dispersion around a VHS point on the $k_{x}-k_{y}$ plane at
$k_{z}=0.62~{}\mathrm{\AA}$. (d) Phonon dispersion along the high symmetry
directions in the bulk BZ. The size and color of markers indicate band-
resolved electron-phonon coupling (EPC) strength. (e) The Eliashberg spectral
function and cumulative EPC at the Fermi energy ($E_{f}$) and the VHS energy
($E_{vhs}=0.4$ eV). The enhancement of electron-phonon interaction due to
peaked DOS at VHS is visible in the Eliashberg spectral function and EPC.

Since RhGe exhibits a superconducting state below $\sim 4.3$ K [48], its
surface states can become superconducting via the proximity effect.
Accordingly, we now investigate the possibility of superconductivity in the
bulk states and discuss the associated pairing symmetries. Figure 3(a) shows
the bulk band structure along the chiral axis ($\Gamma-R$ line) and density of
states (DOS) of RhGe. A careful analysis reveals that a large DOS at 0.4 eV
above the Fermi energy is associated with the VHS. In Fig. 3(b), we illustrate
constant energy contours on the $k_{z}=0.62~{}\mathrm{\AA^{-1}}$ plane. The
crossing of constant energy lines reveals saddle points at symmetry equivalent
generic momentum point $k_{VHS}=(0.62,0.62,0.62)~{}\mathrm{\AA^{-1}}$ on the
$\Gamma-R$ line. The associated saddle-like energy dispersion is shown in Fig.
3(c). Importantly, unlike the usual saddle points, the saddle points here lie
on the chiral axis and on the bands forming the multifold Weyl cones.
Considering the symmetry of the lattice, there are eight such saddle points in
the whole BZ.

Figure 4: (a) Constant energy surface (CES) of RhGe at $E_{vhs}=0.4$ eV. The
color scale represents the strength of electron-phonon coupling. The VHS
located at the [111] chiral axis is marked with a black-red arrow. (b)-(c) The
electronic-phonon coupling weighted CES on (001) and (110) planes. The major
contribution to EPC comes from the pockets around $\Gamma$ point. (d)
Distribution of superconducting gap over the CES. (e)-(f) The superconducting
gap distribution on (001) and (110) plane cuts of the CES. The anisotropic
distribution of EPC and the superconducting gap is evident.

The novel electronic phenomena originating from the saddle-point VHSs can be
probed by tuning the binding energy at the $E_{VHS}=0.4$ eV. This can be
achieved by electron doping as reported in recent experiments [50]. To confirm
the effects of VHS on the enhancement of electron-phonon interaction, we
calculate the phonon spectrum and the momentum resolved EPC
$\lambda_{\mathbf{q}\nu}$, where $\mathbf{q}$ is the wavevector, $\nu$ is the
phonon band index (see SM for details). The EPC-weighted phonon spectrum of
RhGe is shown in Fig. 3(d), which indicates the large contribution of EPC on
an LA branch along the $\Gamma-M$ direction. The phonon DOS, Eliashberg
spectral function, and cumulative EPC display an enhanced EPI strength from
$\lambda_{E_{F}}=0.17$ to $\lambda_{E_{VHS}}=0.78$ as one tune energy from
Fermi level to VHS. This enhancement of EPC strengthens the BCS
superconducting state in RhGe. An estimation of the superconducting transition
temperature using the Allen-Dynes equation,
$T_{c}=\frac{\omega_{\log}}{1.2}\exp\left[\frac{-1.04(1+\lambda)}{\lambda\left(1-0.62\mu^{*}\right)-\mu^{*}}\right],$
with $\lambda_{E_{VHS}}=0.78$ and screened Coulomb interaction parameter
$\mu^{*}=0.1$, is found to be $\approx 5.98$ K, close to experimental reported
$T_{c}$ [48]. Here $\omega_{\log}$ is the logarithmic average of phonon
frequency.

By solving the anisotropic Migdal-Eliashberg equation on the imaginary axis
with 0.25 eV cut-off over the Matsubara frequencies, we further obtain the EPC
and superconducting gap structure over the constant energy surface (CES) at
$E_{VHS}$ (Fig. 4). The calculated EPC and superconducting gap function over
CES are shown in Figs. 4(a) and 4(c), respectively. Both the EPC and
superconducting gap function show anisotropic distribution over the CES in
momentum space. This is more evident from the (001) and (110) cuts presented
in Figs. 4(d)-4(f) and Fig. 4(i)-4(k) for EPC and superconducting gap
function, respectively. Note that without SOC, EPC is the largest for the
pockets centered at the $\Gamma$ point with an anisotropic s-wave-like BCS
superconducting gap distribution on the $k_{x}-k_{y}$ plane and along the
$k_{z}$ direction. A more realistic description of the gap function over the
fermi surface may require calculations with SOC, which is beyond the
computational reach of current first-principles methods.

Therefore, to gain a deeper understanding of the gap function and potential
pairing symmetries influenced by SOC for the Fermi pockets near the $\Gamma$
point, we employ the kinetic model introduced in Ref. [44], see SM for
details. The projected electronic states near the Fermi surface are described
by $\hat{\psi}_{m,\mathbf{k}}$ ($m=3/2,1/2$). These two Fermi pockets are
associated with the multifold fermionic excitations described by the total
angular momentum $S=3/2$ near the $\Gamma$ point. Ignoring the inter-pocket
pairings, the intra-pocket Cooper pairings between the electrons at
$\mathbf{k}$ and $-\mathbf{k}$ are described by the condensate
$b_{m,\mathbf{k}}=\langle\hat{\psi}_{m,\mathbf{k}}\hat{\tilde{\psi}}_{m,\mathbf{k}}\rangle$.
Here, $\hat{\tilde{\psi}}_{m,\mathbf{k}}$ is the time-reversal partner of the
electron states at $\mathbf{k}$:
$\hat{\tilde{\psi}}_{m,\mathbf{k}}=\eta_{m,\mathbf{k}}\hat{\psi}_{m,-\mathbf{k}}$
with $\eta_{m,\mathbf{k}}=e^{2im\phi_{\mathbf{k}}}$, where $\phi_{\mathbf{k}}$
is the azimutual angle of $\mathbf{k}$. Note that the condensate is an even
function of momentum $b_{m,-\mathbf{k}}=b_{m,\mathbf{k}}$ for half-integers
$m=3/2,1/2$, forcing the gap function
$\Delta_{m}(\mathbf{k})=-\sum_{\mathbf{p}}\sum_{m^{\prime}}U_{mm^{\prime}}(\mathbf{k},\mathbf{p})b_{m^{\prime},\mathbf{p}}$
to be even. This can be seen by considering the mean-field theory to decouple
the interactions in the Cooper channel as $H_{\text{pairing
}}=\frac{1}{2}\sum_{\mathbf{k},\mathbf{p}}\sum_{m,m^{\prime}}U_{mm^{\prime}}(\mathbf{k},\mathbf{p})$
$\left(\hat{\tilde{\psi}}_{m,\mathbf{k}}^{\dagger}\hat{\psi}_{m,\mathbf{k}}^{\dagger}b_{m^{\prime},\mathbf{p}}+\hat{\psi}_{m^{\prime},\mathbf{p}}\hat{\tilde{\psi}}_{m^{\prime},\mathbf{p}}b_{m,\mathbf{k}}^{*}-b_{m^{\prime},\mathbf{p}}b_{m,\mathbf{k}}^{*}\right)$
with
$U_{mm^{\prime}}(\mathbf{k},\mathbf{p})=U_{mm^{\prime}}(\mathbf{k},-\mathbf{p})$.
The superconducting gap function
$\Delta_{m}(\mathbf{k})=-\sum_{\mathbf{p}}\sum_{m^{\prime}}U_{mm^{\prime}}(\mathbf{k},\mathbf{p})b_{m^{\prime},\mathbf{p}}$
is clearly even $\Delta_{m}(-\mathbf{k})=\Delta_{m}(\mathbf{k})$. Thus, for
superconducting pairing symmetry, $L$ is an even integer which at the lowest
angular momentum is $s$-wave ($L=0$) and $d$-wave ($L=2$). The nature of
superconducting gap symmetry is determined by the most attractive channel in
expanding the interaction $U_{mm^{\prime}}(\mathbf{k},\mathbf{p})$ in terms of
the basis functions. The $s$-wave pairing is the most symmetric where the gap
function preserves the time-reversal symmetry and yields a trivial
superconducting state. Higher-angular momentum pairings can break the time-
reversal symmetry and give rise to nontrivial superconductivity characterized
by the topological charge $\nu$. Different possible pairing symmetries and
topological states are summarized in Table 1. These results indicate that RhGe
could realize topological superconductivity.

Table 1: Possible superconducting pairing symmetries around $\Gamma$ and the
associated topological states.

$(\mathrm{L},\mathrm{M})$ |  Gap function |  Nature of superconducting state |  Time-reversal symmetry |  Topological charge
---|---|---|---|---
$(0,0)$ |  $s$-wave |  Trivial |  Preserved |  $\nu=0$
$(2,2)$ |  $k_{x}^{2}-k_{y}^{2}+2ik_{x}k_{y}$ |  Weyl |  Broken |  $\nu=+2$
$(2,1)$ |  $\left(k_{x}+ik_{y}\right)k_{z}$ |  Weyl+nodal loop |  Broken |  $\nu=+1$
$(2,0)$ |  $3k_{z}^{2}-k^{2}$ |  nodal loop |  Preserved |  $\nu=0$
$(2,-1)$ |  $\left(k_{x}-ik_{y}\right)k_{z}$ |  Weyl+nodal loop |  Broken |  $\nu=-1$
$(2,-2)$ |  $k_{x}^{2}-k_{y}^{2}-2ik_{x}k_{y}$ |  Weyl |  Broken |  $\nu=-2$

We emphasize that for the pairing symmetries listed in Table 1, we employed a
rotational symmetric effective model. In an actual crystal, however, the
pairing functions follow point-group symmetries. Thus, the effective
interaction in the pairing channel should be expanded in terms of the basis
functions of the corresponding irreducible representations (Fig. 1(d)). The
point group symmetry at $\Gamma$ is described by the double group $T$. The
normal group admits conjugate representations, $E$, with even basis functions
$\left(d_{z^{2}}\equiv 2z^{2}-x^{2}-y^{2},d_{x^{2}-y^{2}}\equiv
x^{2}-y^{2}\right)$. Both pairings would yield nodal-line superconductors. Due
to the conjugate nature of the representation, a phase-locking between them
leads to pairings of types $d_{z^{2}}\pm id_{x^{2}-y^{2}}$, which breaks the
time-reversal symmetry. Since such complex pairings replace the nodal loops
with nodes, they are energetically favored over $d_{z^{2}}$ and
$d_{x^{2}-y^{2}}$ due to the gain in the condensation energy.

Table 2: Possible superconducting pairing symmetries around $R$ and the
associated topological states.

$(\mathrm{L},\mathrm{M})$ |  Gap function |  Nature of superconducting state |  Time-reversal symmetry |  Topological charge
---|---|---|---|---
$(1,1)$ |  $k_{x}+ik_{y}$ |  Weyl |  Broken |  $\nu=+1$
$(1,0)$ |  $k_{z}$ |  Nodal loop |  Preserved |  $\nu=0$
$(1,-1)$ |  $k_{x}-ik_{y}$ |  Weyl |  Broken |  $\nu=-1$

Unlike the $\Gamma$ pocket, the condensate is odd around the $R$ pocket. The
excitations around $R$ are described by a multifold fermion with $S=1$, and
hence $m=1$ for the states near the Fermi surface. The phase factor
$\eta_{m,\mathbf{k}}=\eta_{m,-\mathbf{k}}$ is even and gives rise to an odd
condensate $b_{m,\mathbf{k}}=-b_{m,-\mathbf{k}}$. Therefore, only odd angular
momenta will survive in the expansion of the pairing interaction
$U_{mm^{\prime}}(\mathbf{k},\mathbf{p})$. The possible pairing symmetries are
provided in SMs. The point-group symmetry of $R$ is described by double group
$C_{3}$. The normal group has a one-dimensional representation with the basis
$p_{z}\equiv z$. The corresponding pairing symmetry leads to a nodal-loop
superconductor. The symmetry group also admits a conjugate representation with
basis functions $\left(p_{x}\equiv x,p_{y}\equiv y\right)$. The pairing
symmetry could be $p_{x}\pm ip_{y}$, $i.e.$, the Weyl superconductors (see
Table 2). In recent experiments, RhGe was reported to show superconductivity
with the coexistence of weak ferromagnetism [48], which can substantiate the
topologically non-trivial nature of the superconductivity.

In summary, we have demonstrated RhGe to be a strong candidate for realizing
topological superconductivity. Owing to its structural chirality, RhGe
possesses multifold Weyl fermions located at the $\Gamma$ and $R$ momentum
points on the chiral axis. The bands connecting these momentum points feature
type-II saddle-point VHSs with enhanced EPI and superconductivity. By
performing the anisotropic Eliashberg analysis, we demonstrate that both the
EPC and the BCS gap exhibit pronounced anisotropy across the Fermi surface. On
characterizing the pairing symmetries around the $\Gamma$ and $R$ points using
a low-energy model, we establish the nontrivial superconducting state in RhGe.
Moreover, the lack of inversion symmetry in RhGe can allow the development of
an admixture of pairing symmetries and strengthen the unconventional
superconducting state. The presence of time-reversal-symmetry-broken
topological superconductivity can be verified via future magnetic optical Kerr
and $\mu$-SR experiments. Our study establishes that RhGe is a promising B20
Weyl semimetal for exploring topological superconductivity and Majorana
fermions.

Acknowledgements. This work is supported by the Department of Atomic Energy of
the Government of India under Project No. 12-R$\&$D-TFR-5.10-0100 and
benefited from the computational resources of TIFR Mumbai. H.L. acknowledges
the support by the National Science and Technology Council (NSTC) in Taiwan
under grant number MOST 111-2112-M-001-057-MY3. The work at Howard University
was supported by the US Department of Energy, Office of Science, Basic Energy
Sciences Grant No. DE-SC0022216. The work at Northeastern University was
supported by the US Department of Energy (DOE), Office of Science, Basic
Energy Sciences Grant No. DE-SC0022216 and benefited from Northeastern
University’s Advanced Scientific Computation Center and the Discovery Cluster,
and the National Energy Research Scientific Computing Center through DOE Grant
No. DE-AC02-05CH11231. The authors acknowledge the Texas Advanced Computing
Center (TACC) at The University of Texas at Austin for providing HPC resources
that have contributed to the research results reported in this paper.

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Existence, Regularity, and a Strong Itô Formula for the Isochronal Phase of
SPDE

(preprint)

Zachary P. Adams

Max Planck Institute for Mathematics in the Sciences

<EMAIL_ADDRESS>

Abstract

We prove the existence and regularity of the isochron map for stable invariant
manifolds of a large class of evolution equations. Our results apply in
particular to the isochron map of reaction-diffusion equations and neural
field equations. Using the regularity properties proven here, we are able to
obtain a strong Itô formula for the isochronal phase of stochastically
perturbed travelling waves, spiral waves, and other patterns appearing in
SPDEs driven by white noise, even for SPDEs that only admit mild solutions.

_Keywords_ :  Isochronal phase $\cdot$ Invariant manifold $\cdot$ Strong Itô
formula $\cdot$ Reaction-diffusion equations $\cdot$ Neural field equations

##  1 Introduction

This note presents simple arguments on the existence and regularity of the
isochron map for a general dynamical system $(\phi_{t})_{t\geq 0}$ on a
separable Banach space in the vicinity of a stable, normally hyperbolic
invariant manifold $\Gamma$. The isochron map was introduced for stable limit
cycles of dynamical systems in $\mathbb{R}^{d}$ by Winfree [31] and
Gukenheimer [12]. Our framework encompasses infinite dimensional dynamical
systems, in particular reaction-diffusion systems and neural field equations.
The regularity properties of the isochron map that we prove allow us to prove
a strong Itô formula for the isochronal phase of stochastic perturbations of
reaction-diffusion systems, even when these systems only admit a mild solution
theory.

The proofs in this note are short and general, and recover results on the
isochron map proven elsewhere in the literature using different techniques
[1]. As will be seen, while [1] requires the invariant manifold to be smooth,
our arguments only require $\Gamma$ to be $C^{1}$. Additionally, we make no
assumptions on the spectral properties of the generator of the linearization
of $\phi_{t}$ about $\Gamma$. This is of particular interest in the case where
$\Gamma$ is the orbit of a non-commutative Lie group (such as $SE(n)$ when
$\Gamma$ consists of spiral wave solutions to a parabolic PDE), where trying
to track the linearized dynamics along $\Gamma$ can lead to significant
complications, as noted in the introduction of [23].

As a consequence, the approach taken in this note may lead to significant
generalizations of results obtained for travelling pulses and other transient
patterns appearing in stochastic evolution equations using other notions of
“phase”. For instance, those studied in neural field equations by Inglis &
MacLaurin [19], Lang [24], and Lang & Stannat [25], and in reaction-diffusion
equations by Eichinger, Gnann, & Kuehn [11], Hamster & Hupkes [15, 16, 17],
Krüger & Stannat [21, 22], and Stannat [29].

##  2 Setup

We consider an evolution equation of the form

$\partial_{t}x\,=\,Lx+N(x)\,\eqqcolon\,V(x),$ (1)

and its stochastic perturbation

$dX\,=\,\left(LX+N(X)\right)\,dt+\sigma B\,dW.$ (2)

Here, $L$ is a linear operator generating a $C_{0}$-semigroup on a Hilbert
space $H$, $N$ is a nonlinearity defined on a densely, continuously embedded
Banach subspace $E$ of $H$, $\sigma\geq 0$, $B$ is a linear operator on $H$,
and $(W_{t})_{t\geq 0}$ is a cylindrical Wiener process, as defined in [10,
Chapter 4.1.2]. More precisely, we make the following assumptions on (1) and
(2).

######  Assumption 1.

The system (1) satisfies the following.

  1. (a)

$N$ is a nonlinearity defined on some $E\subset H$ such that $E$ is a Banach
space with norm $\left\|\cdot\right\|_{E}$, and the embedding
$E\hookrightarrow H$ is continuous. We moreover assume that $N$ is four times
continuously Fréchet differentiable in the topology of $E$, and denote by $DN$
and $D^{2}N$ its first and second Fréchet derivatives in this topology.

  2. (b)

$L$ is a linear operator with a dense domain of definition $D(L)$ in $E$, and
generates a $C_{0}$-semigroup $(\Lambda_{t})_{t\geq 0}$ on $H$. Moreover,
letting $\left\|\cdot\right\|$ denote the norm of $H$, there exists $\omega>0$
such that for all $t\geq 0$,

$\sup\left\\{\left\|\Lambda_{t}\right\|,\,\left\|\Lambda_{t}\right\|_{E}\right\\}\,\leq\,e^{-\omega
t}.$ (3)

  3. (c)

There exists an orthonormal basis $\\{e_{k}\\}_{k\in\mathbb{N}}$ of $H$
consisting of eigenfunctions of $L$. For all $r>s>0$ and $k\in\mathbb{N}$, we
have $\Lambda_{r}e_{k}\in E$ and

$K_{s,r}\,\coloneqq\,\sup_{t\in[s,r]}\sum_{k\in\mathbb{N}}\left\|\Lambda_{t}e_{k}\right\|_{E}\,<\,\infty.$
(4)

Additionally, letting $\lambda_{k}$ denote the eigenvalue of $-L$
corresponding to $e_{k}$, there exists $C_{\lambda}>0$ such that, uniformly in
$k\in\mathbb{N}$, we have

$\left(\lambda_{k+1}-\lambda_{k}\right)^{-1}\ln\frac{\lambda_{k+1}}{\lambda_{k}}\,\leq\,C_{\lambda}.$
(5)

Assumption 1(c) is only needed when the noise in (2) is white in space, but
not when the noise is trace class. Let the flow map of (1) be
$(t,x)\,\mapsto\,\phi_{t}(x)$, and suppose that the flow can be expressed by
the variation of constants formula as

$\phi_{t}(x)\,=\,\Lambda_{t}x+\int_{0}^{t}\Lambda_{t-s}N(\phi_{s}(x))\,ds.$

Since the nonlinearity $N$ is $C^{4}$ in the topology of $E$, the flow map is
three times continuously Fréchet differentiable in the topology of $E$. The
first, second, and third derivatives of $\phi_{t}$ at $x_{0}$ and in
directions $x,y,z\in E$ are denoted by

$x\,\mapsto\,D\phi_{t}(x_{0})[x],\qquad(x,y)\,\mapsto\,D^{2}\phi_{t}(x_{0})[x,y],\qquad(x,y,z)\,\mapsto\,D^{3}\phi_{t}(x_{0})[x,y,z].$
(6)

We now state our assumptions on the flow of (1). Specifically, we assume the
existence of a stable invariant manifold of (1) that models spatiotemporal
patterns.

######  Assumption 2.

The deterministic system (1) has a stable normally hyperbolic invariant
manifold $\Gamma$, as defined in [3, Condition (H3)], that can be
parameterized by a manifold $\mathcal{S}\subset\mathbb{R}^{m}$, where
$m\in\mathbb{N}$. We write
$\Gamma=\\{\gamma_{\alpha}\\}_{\alpha\in\mathcal{S}}$, and let $B(\Gamma)$
denote the basin of attraction of $\Gamma$ in $H$. Moreover, the following
hold.

  1. (a)

$\phi_{t}$ is bi-Lipschitz on $\Gamma$ with uniform in time Lipschitz
constants. Specifically, there exists $C>0$ such that for all $t>0$ and
$x,y\in\Gamma$,

$\left\|\phi_{t}(x)-\phi_{t}(y)\right\|_{E}\,\leq\,C\left\|x-y\right\|_{E},\qquad\left\|\phi_{t}^{-1}(x)-\phi_{t}^{-1}(y)\right\|_{E}\,\leq\,C\left\|x-y\right\|_{E},$

  2. (b)

$D\phi_{t}(\gamma_{\alpha})$ is invertible with uniformly bounded inverse for
all $\alpha\in\mathcal{S}$.

  3. (c)

The map $\alpha\mapsto\gamma_{\alpha}$ is continuously differentiable with
derivative $D\gamma_{\alpha}$, and $D\gamma_{\alpha}$ is invertible with a
uniformly bounded inverse.

######  Remark 2.1.

We denote the tangent space of $\mathcal{S}$ at $\alpha\in\mathcal{S}$ by
$T_{\alpha}\mathcal{S}$, and the tangent bundle of $\mathcal{S}$ by
$T\mathcal{S}$ – see [18] for definitions. Similarly, the cotangent space of
$\mathcal{S}$ at $\alpha\in\mathcal{S}$ is denoted
$T_{\alpha}^{*}\mathcal{S}$, and we write the action of $u^{*}\in
T_{\alpha}^{*}\mathcal{S}$ on $u\in T_{\alpha}\mathcal{S}$ as $\left\langle
u^{*},u\right\rangle$. If the intrinsic dimension of $\mathcal{S}$ is
$m^{\prime}\leq m$, note that we can identify each $T_{\alpha}\mathcal{S}$ and
$T_{\alpha}^{*}\mathcal{S}$ with an $m^{\prime}$-dimensional subspace of
$\mathbb{R}^{m}$, and as the embedding
$\mathcal{S}\hookrightarrow\mathbb{R}^{m}$ is fixed, this identification is
unique. By Assumption 2, the covariant derivative of a function $g$ on
$\mathcal{S}$ at $\alpha\in\mathcal{S}$ in a direction $v\in
T_{\alpha}\mathcal{S}$ is equivalent to a directional derivative of $g$ taken
with respect to the topology of $\mathbb{R}^{m}$. Hence, we write the
derivative of a function $g\in C^{k}(\mathcal{S})$ at $\alpha\in\mathcal{S}$
in the direction $v\in T_{\alpha}\mathcal{S}$ as $Dg(\alpha)v$.

The bi-Lipschitz condition above ensures that the dynamics of (1) is not
chaotic on $\Gamma$. Assumption 2 is used to prove the existence and
regularity of the isochron map in Section 1.

######  Definition 2.2.

Assumption 2 implies that for each $x\in B(\Gamma)$ there is a unique
$\pi(x)\in\mathcal{S}$ such that

$\left\|\phi_{t}(x)-\phi_{t}(\gamma_{\pi(x)})\right\|\,\xrightarrow[t\rightarrow\infty]{}\,0,$
(7)

providing a well-defined map $\pi:B(\Gamma)\rightarrow\mathcal{S}$. The
existence and uniqueness of this map is proven in Theorem 3.1. We refer to
$\pi$ as the _isochron map_ of $\Gamma$, in analogy to the isochron map
introduced by Winfree [31] and Guckenheimer [12]. If (2) has a well-posed
solution $(X_{t})_{t\geq 0}$ such that $X_{t}\in B(\Gamma)$ for some $t>0$, we
refer to $\pi(X_{t})$ as the _isochronal phase_ of $X_{t}$.

Throughout this note, we let $\Gamma_{\delta}$ denote the following
$\delta$-neighbourhood of $\Gamma$,

$\Gamma_{\delta}\,\coloneqq\,\left\\{x\in
E\,:\,\left\|x-\gamma_{\pi(x)}\right\|_{E}\leq\delta\right\\}$ (8)

for $\delta>0$. Since we define $\Gamma_{\delta}$ in the topology of $E$, the
map $x\mapsto\phi_{t}(x)$ is three times Fréchet differentiable in the
topology of $E$ at any $x\in\Gamma_{\delta}$.

Now, as noted in the introduction, we are concerned with stochastic
perturbations of (1). The noise in (2) is defined to be a cylindrical Wiener
process $W$ on $H$, so that it is uncorrelated in space and time. The case of
spatially correlated noise can be treated by adding correlation via the
diffusion coefficient $B$. We make the following assumptions.

######  Assumption 3.

The SPDE (2) satisfies the following.

  1. (a)

There exists a unique $E$-valued mild solution $(X_{t})_{t\geq 0}$ to (2),
satisfying

$X_{t}\,=\,\Lambda_{t}X_{0}+\int_{0}^{t}\Lambda_{t-s}N(X_{s})\,ds+\sigma\int_{0}^{t}\Lambda_{t-s}B\,dW_{s},\qquad
X_{0}\in H,$ (9)

for all $t<\tau_{blowup}$, where $\tau_{blowup}$ is an almost surely positive
stopping time possibly equal to infinity.

  2. (b)

For all $t<\tau_{blowup}$,

$\lim_{h\rightarrow
0}\sup_{s\in[0,t]}h^{-1/2}\left\|(\Lambda_{h}-I)X_{s}\right\|_{E}\,=\,0.$ (10)

Note that with additive noise, solutions of (2) may be pushed out of
$B(\Gamma)$ in finite time. We therefore define the exit time of the solution
from $B(\Gamma)$ as

$\tau\,\coloneqq\,\min\left\\{\tau_{blowup},\,\inf\left\\{t>0\,:\,X_{t}\in
E/B(\Gamma)\right\\}\right\\}.$ (11)

Since the geometry of $B(\Gamma)$ is generally unknown and may be very
complicated, we will sometimes restrict our attention to $\Gamma_{\delta}$ for
$\delta>0$. It will prove useful to define the neighbourhood $\Gamma_{\delta}$
in terms of an “isochronal distance”. Hence, we define the exit time from
$\Gamma_{\delta}$ as

$\tau_{\delta}\,\coloneqq\,\min\left\\{\tau_{blowup},\,\inf\left\\{t>0\,:\,\left\|X_{t}-\gamma_{\pi(X_{t})}\right\|_{E}=\delta\right\\}\right\\}.$
(12)

We let $\pi_{t}=\pi(X_{t})$ be the isochronal phase of $X_{t}$ for $t<\tau$.

######  Remark 2.3.

Two classes of examples of (1) motivate the present discussion. The first is
the class of reaction-diffusion systems of the form

$\partial_{t}x\,=\,(\Delta-a)x+N(x),$ (13)

where $a>0$ and $N$ is a vector of polynomials. With $L=\Delta-a$ and $O$
bounded, we can show that (13) satisfies Assumption 1. For instance if
$O=[0,\ell]$ with Dirichlet or periodic boundaries, take $H=L^{2}(O)$ and
$E=C_{b}(O)$. We may take $e_{k}(x)=\sin\left(\frac{k\pi}{\ell}x\right)$ and
$\lambda_{k}=\frac{\pi^{2}k^{2}}{\ell^{2}}$, so that (5) is satisfied.
Moreover, note that for $t>0$ there exists $c(t)>0$ independent of $k$ and
increasing in $t$ such that

$\left\|\Lambda_{t}e_{k}\right\|_{E}\,=\,\exp\left(-\frac{\pi^{2}t}{\ell^{2}}k^{2}-at\right)\left\|e_{k}\right\|_{E}\,<\,c(t)k^{-2}.$

Therefore for fixed $r>s>0$ we have

$K_{s,r}\,=\,\sup_{t\in[s,r]}\sum_{k\in\mathbb{N}}\left\|\Lambda_{t}e_{k}\right\|_{E}\,\leq\,c(s)\sum_{k\in\mathbb{N}}k^{-2}\,<\,\infty.$

The condition (5) is similarly verified. When such a reaction-diffusion system
is perturbed by additive white noise, Assumption 3 is satisfied by Proposition
A.1.

The second class of examples in which we are interested in is a class of
integro-differential equations, with dynamics at $\xi\in O$ described as

$\displaystyle\partial_{t}x(t,\xi)$
$\displaystyle=\,-x(t,\xi)+\int_{O}\omega(\xi,\zeta)f(x(t,\zeta))\,d\zeta,$
(14) $\displaystyle\partial_{t}y(t,\xi)$
$\displaystyle=\,-\varepsilon^{-1}y(t,\xi)+x(t,\xi).$

where $\omega$ and $f$ are typically bounded and Lipschitz. These equations
are frequently employed in neuroscience (where they are typically referred to
as _neural field equations_) [7] and ecology [28]. Mild solutions to
stochastic perturbations of (14) only make sense when the perturbing noise is
trace class [22], and in this case Assumption 3 is satisfied. With
$L=[-1,-\epsilon^{-1}]$, Assumptions 1(a), (b) are satisfied, though (c) is
not. However, Assumption 1 (c) is only needed to prove the Itô formula for the
isochronal phase when $W$ is not trace class, as can be seen in the arguments
of Section 3 below. As stochastic neural field equations only make sense for
trace class noise, the fact that (14) does not satisfy Assumption 1(c) is not
an issue.

For either reaction-diffusion or neural field equations, the principal
examples of $\Gamma$ that we have in mind are stationary or travelling waves
of activity in neural field equations or nerve axon equations [2, 20, 26], and
spiral waves in excitable media, as studied by Beyn & Thümmler [5, 6] on
unbounded domains and by Xin [32] on a bounded spatial domain – see also [23].
We are also interested in the case of stationary patterns that remain
stationary under translation or rotation, as occur in both reaction-diffusion
and neural field equations [8, 26, 30]. So long as either of (13) or (14)
possesses a sufficiently regular invariant manifold $\Gamma$, and the
parameterization $\alpha\mapsto\gamma_{\alpha}$ is non-degenerate, Assumption
2 is satisfied. For instance, if $\Gamma$ consists of travelling wave
solutions on a periodic domain, this is the case.

##  3 Regularity of the Isochron Map & a Strong Itô Formula

In this section we prove regularity properties of the isochron map, along with
an Itô formula for the isochronal phase $\pi(X_{t})$ of the SPDE (2). We begin
with the following result.

######  Theorem 3.1.

Under Assumptions 1 & 2 hold, the function
$\pi:B(\Gamma)\rightarrow\mathcal{S}$ defined in (7) is well-defined and twice
continuously Fréchet differentiable at each $x_{0}\in B(\Gamma)\subset E$ in
the topology of $E$.

###### Proof.

We begin by proving that $\pi:B(\Gamma)\rightarrow\mathcal{S}$ is a well-
defined map. Since $\Gamma$ is a stable manifold of (1), taking any sequence
of positive numbers $\\{\epsilon_{n}\\}_{n\in\mathbb{N}}$ monotonically
decreasing to zero, there exists a sequence of times
$\\{t_{n}\\}_{n\in\mathbb{N}}$ monotonically increasing to infinity and a
sequence of closed sets $U_{n}\subset\Gamma$ such that for all
$n\in\mathbb{N}$ and $\gamma\in U_{n}$, we have

$\left\|\phi_{t_{n}}(x)-\gamma\right\|_{E}\,\leq\,\epsilon_{n}/2M_{1},$ (15)

for some constant $M_{1}$. It can be seen that
$\\{\phi_{t_{n}}^{-1}U_{n}\\}_{n\in\mathbb{N}}$ constitutes a sequence of non-
empty, closed, nested subsets of the complete metric space $E$. Moreover,
using the bi-Lipschitz condition of $\phi_{t}$ on $\Gamma$ and (15), we see
that

$\operatorname{diam}\left(\phi_{t_{n}}^{-1}U_{n}\right)\,\leq\,\epsilon_{n}.$

In particular, the diameter of $\phi_{t_{n}}^{-1}U_{n}$ tends to zero as
$n\rightarrow\infty$. By Cantor’s Intersection Theorem, it follows that there
is a unique $\gamma_{*}$ such that

$\bigcap_{n\in\mathbb{N}}\phi_{t_{n}}^{-1}U_{n}\,=\,\\{\gamma_{*}\\}.$

Since for each $\gamma\in\Gamma$ we have assumed that there is a unique
$\alpha\in\mathcal{S}$ such that $\gamma=\gamma_{\alpha}$, there is a unique
$\pi(x)\in\mathcal{S}$ such that $\gamma_{*}=\gamma_{\pi(x)}$.

To prove that $\pi$ is $C^{2}$, recall that since $N$ is $C^{4}$ by assumption
the map $x\mapsto\phi_{t}(x)$ is $C^{3}$ on $B(\Gamma)\cap E$ in the topology
of $E$. Since $C^{3}$ is a Banach space, the map

$x\,\mapsto\,\gamma_{\pi(x)}\,=\,\lim_{t\rightarrow\infty}\phi_{t}(x)$

is $C^{3}$ on $B(\Gamma)\cap E$. By assumption,
$(r\mapsto\gamma_{r}):\mathcal{S}\rightarrow E$ is an invertible map with
$C^{2}$ inverse. Denoting this inverse by $\gamma^{-1}$, the map

$x\,\mapsto\,\pi(x)\,=\,\gamma^{-1}(\gamma_{\pi(x)})$

is $C^{2}$ on $B(\Gamma)\cap E$ in the topology of $E$. ∎

We henceforth denote the first derivative of the isochron map at $x_{0}\in
B(\Gamma)\cap E$ in the direction $x\in E$ by $D\pi(x_{0})x$. Similarly, the
second derivative of $\pi$ at $x_{0}\in B(\Gamma)\cap E$ in the directions
$x,y\in E$ is denoted $D^{2}\pi(x_{0})[x,y]$. The following theorem provides
additional regularity of the isochron map when Assumption 1(c) holds, showing
in particular that its second derivative is trace class, in a certain sense.

######  Theorem 3.2.

Under Assumptions 1 & 2, there exists $M_{\pi}\in(0,\infty)$ such that

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D^{2}\pi(x_{0})[e_{k},e_{k}]\right\|_{\mathbb{R}^{m}}$
$\displaystyle<\,M_{\pi},$ (16)
$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\pi(x_{0})e_{k}\right\|_{\mathbb{R}^{m}}$
$\displaystyle<\,M_{\pi},$
$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\pi(x_{0})e_{k}\right\|_{\mathbb{R}^{m}}^{2}$
$\displaystyle<\,M_{\pi},$

uniformly in $x_{0}\in\Gamma_{\delta}$. Moreover, the maps defined by sending
$x_{0}\in\Gamma_{\delta}$ to the quantities in (16) are Lipschitz continuous
in $\Gamma_{\delta}$.

To prove Theorem 3.2, define $\tilde{\pi}:\Gamma_{\delta}\rightarrow\Gamma$ as
$\tilde{\pi}(x)\coloneqq\gamma_{\pi(x)}$, and note that

$\phi_{t}(\tilde{\pi}(x))\,=\,\tilde{\pi}(\phi_{t}(x))\qquad\forall
x\in\Gamma_{\delta},\,\,\,t>0.$ (17)

We show that (16) is satisfied by $\pi$ replaced with $\tilde{\pi}$. Then,
since the map $\alpha\mapsto\gamma_{\alpha}$ is $C^{2}$ with uniformly bounded
first derivative, (16) follows. From (17) and Assumption 2, we have

$D\tilde{\pi}(x)y\,=\,[D\phi_{t}(\tilde{\pi}(x))]^{-1}D\tilde{\pi}(\phi_{t}(x))D\phi_{t}(x)y,$
(18)

and

$\displaystyle D^{2}\tilde{\pi}(x)[y,z]$
$\displaystyle=\,[D\phi_{t}(\tilde{\pi}(x))]^{-1}\bigg{(}D^{2}\tilde{\pi}(\phi_{t}(x))[D\phi_{t}(x)y,D\phi_{t}(x)z]$
(19)
$\displaystyle\qquad+D\tilde{\pi}(\phi_{t}(x))D^{2}\phi_{t}(x)[y,z]-D^{2}\phi_{t}(\tilde{\pi}(x))[D\tilde{\pi}(x)y,D\tilde{\pi}(x)z]\bigg{)}.$

We now let $\left\|\cdot\right\|_{1}\coloneqq\left\|\right\|_{E\rightarrow E}$
and $\left\|\cdot\right\|_{2}\coloneqq\left\|\cdot\right\|_{E\times
E\rightarrow E}$. Since $D\phi_{t}(x)^{-1}$ and $D\tilde{\pi}(\phi_{t}(x))$
are bounded operators, it follows that for some $C>0$ independent of $x\in
B(\Gamma)$ and $t>0$,

$\displaystyle\sum_{k\in\mathbb{N}}\|D\tilde{\pi}(x)e_{k}\|_{E}$
$\displaystyle\leq\,\left\|D\phi_{t}(\tilde{\pi}(x))^{-1}D\tilde{\pi}(\phi_{t}(x))\right\|_{1}\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x)e_{k}\right\|_{E}$
(20)
$\displaystyle\leq\,C\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x)e_{k}\right\|_{E}.$

Similarly, since $D\phi_{t}(\tilde{\pi}(x))^{-1}$,
$D^{2}\tilde{\pi}(\phi_{t}(x))$, $D\tilde{\pi}(\phi_{t}(x))$, and
$D^{2}\phi_{t}(x)$ are bounded uniformly in $x\in\Gamma_{\delta}$, for some
$C_{1},C_{2},C_{3}>0$ independent of $x\in B(\Gamma)$ and $t>0$,

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D^{2}\tilde{\pi}(x)[e_{k},e_{k}]\right\|_{1}$
$\displaystyle\leq\,\left\|D\phi_{t}(\tilde{\pi}(x))^{-1}\right\|_{E}\bigg{(}\left\|D^{2}\tilde{\pi}(\phi_{t}(x))\right\|_{2}\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x)e_{k}\right\|_{E}^{2}$
(21)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\left\|D\tilde{\pi}_{E}(\phi_{t}(x))\right\|_{1}\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{t}(x)[e_{k},e_{k}]\right\|_{E}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\left\|D^{2}\phi_{t}(\tilde{\pi}(x))\right\|_{2}\sum_{k\in\mathbb{N}}\left\|D\tilde{\pi}(x)e_{k}\right\|_{E}^{2}\bigg{)}$
$\displaystyle\leq\,C_{1}\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x)e_{k}\right\|_{E}^{2}+C_{2}\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{t}(x)[e_{k},e_{k}]\right\|_{E}+C_{3}\sum_{k\in\mathbb{N}}\left\|D\tilde{\pi}(x)e_{k}\right\|^{2}_{E}.$

Hence, Theorem 3.2 immediately follows as a corollary to the following result.

######  Lemma 3.3.

Under Assumptions 1 & 2, there exists $M_{\phi}\in(0,\infty)$ such that for
any $t>0$ and small $\delta>0$,

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,<\,M_{\phi},$
(22)
$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}^{2}\,<\,M_{\phi},$
(23)
$\displaystyle\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{s}(x_{0})[e_{k},e_{k}]\right\|_{E}\,<\,M_{\phi},$
(24)

uniformly in $x_{0}\in\Gamma_{\delta}$ and $s\in(0,t)$. Moreover, the maps
defined by sending $x_{0}$ to the values defined in (22), (23), (24) are
Lipschitz continuous on $\Gamma_{\delta}$.

###### Proof.

We consider the evolution equations of the operators defined in (6), written
in mild form as

$\displaystyle
D\phi_{t}(x_{0})[x]\,=\Lambda_{t}D\phi_{0}(x_{0})[x]+\int_{0}^{t}\Lambda_{t-s}DN(\phi_{s}(x_{0}))D\phi_{s}(x_{0})[x]\,ds,$
(25) $\displaystyle
D^{2}\phi_{t}(x_{0})[x,y]\,=\,\Lambda_{t}D^{2}\phi_{0}(x_{0})[x,y]+\int_{0}^{t}\Lambda_{t-s}DN(\phi_{s}(x_{0}))D^{2}\phi_{s}(x_{0})[x,y]\,ds$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{0}^{t}\Lambda_{t-s}D^{2}N(\phi_{s}(x_{0}))\big{[}D\phi_{s}(x_{0})[x],D\phi_{s}(x_{0})[y]\big{]}\,ds,$

where $DN$ and $D^{2}N$ are the first and second Fréchet derivatives of $N$.

Now, we observe that $D\phi_{0}(x_{0})[x]=x$, so

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}$
$\displaystyle\leq\,\sum_{k\in\mathbb{N}}\left\|\Lambda_{t}e_{k}\right\|_{E}+\sum_{k\in\mathbb{N}}\int_{0}^{t}\left\|\Lambda_{t-s}DN(\phi_{s}(x_{0}))\right\|_{1}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds$
$\displaystyle\leq\,\sum_{k\in\mathbb{N}}\left\|\Lambda_{t}e_{k}\right\|_{E}+C_{0}\int_{0}^{t}\sum_{k\in\mathbb{N}}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds$

for some $C_{0}>0$ which is independent of $x_{0}$ (using (3), and the fact
that $N$ is uniformly Lipschitz on $\Gamma_{\delta}$ in the topology of $E$).
Taking $s\in(0,t)$ and letting $r\mapsto K_{s,r}$ be the non-decreasing
function defined in Assumption C, we apply Grönwall’s inequality to obtain,
for arbitrary $r>t$,

$\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}\,\leq\,K_{s,r}e^{C_{0}t}\,<\,\infty.$
(26)

Furthermore, fixing $s<t<r$, (26) implies a local integrability of the sum,

$\sum_{k\in\mathbb{N}}\int_{0}^{t}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds\,=\,\int_{0}^{t}\sum_{k\in\mathbb{N}}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds\,<\,\frac{K_{s,r}}{C_{0}}e^{C_{0}t}.$

We remark that the constants appearing in this bound are independent of
$x_{0}\in\Gamma_{\delta}$, implying the uniform bound (22).

Similarly, observe that

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}^{2}$
$\displaystyle\leq\,\sum_{k\in\mathbb{N}}\left\|\Lambda_{t}e_{k}\right\|_{E}^{2}+2C_{0}\sum_{k\in\mathbb{N}}\left(\left\|\Lambda_{t}e_{k}\right\|_{E}\int_{0}^{t}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds\right)$
$\displaystyle\,\,\,+C_{0}^{2}\sum_{k\in\mathbb{N}}\left(\int_{0}^{t}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds\right)^{2}.$

The first sum is bounded, by (4). The second sum is bounded due to (26) and
the fact that for any two summable sequences of positive numbers
$(a_{k})_{k\in\mathbb{N}},\,(b_{k})_{k\in\mathbb{N}}$ we have

$\sum_{k\in\mathbb{N}}a_{k}b_{k}\,\leq\,\sum_{k\in\mathbb{N}}a_{k}b_{k}+\sum_{k\neq\ell}a_{k}b_{\ell}\,=\,\left(\sum_{k\in\mathbb{N}}a_{k}\right)\left(\sum_{k\in\mathbb{N}}b_{k}\right).$

Therefore, for arbitrary $s<t<r$,

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}^{2}$
$\displaystyle\leq\,K_{s,r}^{2}+2K_{s,r}^{2}e^{C_{0}t}+C_{0}^{2}\sum_{k\in\mathbb{N}}\left(\int_{0}^{t}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}\,ds\right)^{2}$
$\displaystyle\leq\,K_{s,r}^{2}\left(1+2e^{C_{0}t}\right)+C_{0}^{2}\int_{0}^{t}\sum_{k\in\mathbb{N}}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}^{2}\,ds.$

Applying Grönwall’s inequality again, we find that
$\sum_{k\in\mathbb{N}}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}^{2}$ is
finite and locally integrable in $t>0$. We remark that the constants appearing
in the bound of (23) obtained by these arguments are independent of $x_{0}$.

Once we have the uniform bounds (22) & (23), we can similarly obtain (24).
Since $D^{2}\phi_{0}(x_{0})[x,x]=x$, we have

$\displaystyle\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{t}(x_{0})[e_{k},e_{k}]\right\|_{E}$
$\displaystyle\leq\,\sum_{k\in\mathbb{N}}\left\|\Lambda_{t}e_{k}\right\|_{E}+C_{1}\int_{0}^{t}\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{s}(x_{0})[e_{k},e_{k}]\right\|_{E}\,ds$
$\displaystyle\qquad\qquad\qquad\qquad+C_{1}\int_{0}^{t}\sum_{k\in\mathbb{N}}\left\|D\phi_{s}(x_{0})[e_{k}]\right\|_{E}^{2}\,ds.$

By our previous results, the first and third sum in the above are finite.
Therefore for arbitrary $s<t<r$ we have

$\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{t}(x_{0})[e_{k},e_{k}]\right\|_{E}\,\leq\,L_{s,r}+C\int_{0}^{t}\sum_{k\in\mathbb{N}}\left\|D^{2}\phi_{s}(x_{0})[e_{k},e_{k}]\right\|_{E}\,ds$

for another constant $L_{s,r}>0$. We may therefore apply Grönwall’s inequality
again, to obtain the final bound (24) with constants independent of $x_{0}$.

To see Lipschitz continuity of the maps defined by taking $x_{0}$ to the
values in (22), (23), (24), we remark that the first part of this proposition
implies that the finite sums

$\displaystyle\sum_{k=1}^{K}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}\,<\,M_{\phi},$
$\displaystyle\sum_{k=1}^{K}\left\|D\phi_{t}(x_{0})[e_{k}]\right\|_{E}^{2}\,<\,M_{\phi},$
$\displaystyle\sum_{k=1}^{K}\left\|D^{2}\phi_{t}(x_{0})[e_{k},e_{k}]\right\|_{E}\,<\,M_{\phi},$

converge uniformly in $x_{0}\in\Gamma_{\delta}$ to (22), (23), (24) as
$K\rightarrow\infty$. Hence we need only prove that the finite sums are
Lipschitz. Indeed, from the fact that $x_{0}\mapsto\phi_{t}(x_{0})$ is $C^{3}$
in the topology of $E$ (since $N$ is $C^{4}$), we know that each of the
summands is Lipschitz in $\Gamma_{\delta}$. Therefore, the finite sums are
Lipschitz. This completes the proof. ∎

We now prove a strong Itô formula for $\pi(X_{t})$ that holds for $t<\tau$.
Note that in the proof of Theorem 3.6, since we have shown above that

$\sum_{k\in\mathbb{N}}\left\|D^{2}\pi(x_{0})[e_{k},e_{k}]\right\|_{\mathbb{R}^{m}}\,<\,\infty$

for any $x_{0}\in B(\Gamma)\cap E$, we do not need to assume that $B$ is trace
class. In particular, we are able to take $W$ as an uncorrelated cylindrical
Wiener process. As our goal is to prove a strong Itô formula for $\pi(X_{t})$,
we need the following two lemmas.

######  Lemma 3.4.

Let Assumptions 1, 2, & 3 hold. For each $t>0$, the Itô integral

$\int_{0}^{t\wedge\tau}D\pi(X_{s})B\,dW_{s}$ (27)

is well-defined as an element of $\mathbb{R}^{m}$.

###### Proof.

We prove the result for noise that is white in space and time. Modifying the
proof to apply to coloured noise is trivial. For a collection of independent
identically distributed Brownian motions $\\{\beta^{k}\\}_{k\in\mathbb{N}}$,
write

$W_{t}=\sum_{k\in\mathbb{N}}e_{k}\beta^{k}_{t}.$ (28)

Of course, the sum (28) does not converge in any sense as an $H$-valued random
variable, but as an $H_{0}$-valued random variable for some Hilbert space
$H_{0}\supset H$ (see Chapter 4 of Da Prato & Zabczyk [10] for details).
However, we need not specify $H_{0}$ here. Indeed, the integral (27) can be
made sense of as the limit in expectation of the finite sums

$\sum_{k=1}^{K}\int_{0}^{t\wedge\tau}D\pi(X_{s})Be_{k}\,d\beta_{s}^{k}\,\eqqcolon\,\sum_{k=1}^{K}\tilde{\beta}_{t}^{k},$

which converge in $\mathbb{R}^{m}$. Using Assumption E, Theorem 3.2, and Itô’s
isometry, we obtain

$\displaystyle\operatorname{\mathbb{E}}\left[\left\|\sum_{k\in\mathbb{N}}\tilde{\beta}_{t}^{k}\right\|_{\mathbb{R}^{m}}\right]^{2}$
$\displaystyle\leq\,\sum_{k\in\mathbb{N}}\operatorname{\mathbb{E}}\left[\left\|\tilde{\beta}_{t}^{k}\right\|_{\mathbb{R}^{m}}^{2}\right]+\sum_{k\neq\ell}\operatorname{\mathbb{E}}\left[\left\|\tilde{\beta}_{t}^{k}\right\|_{\mathbb{R}^{m}}\left\|\tilde{\beta}_{t}^{\ell}\right\|_{\mathbb{R}^{m}}\right]$
(29)
$\displaystyle\leq\,\sum_{k\in\mathbb{N}}\operatorname{\mathbb{E}}\left[\int_{0}^{t\wedge\tau}\left\|D\pi(X_{s})Be_{k}\right\|_{\mathbb{R}^{m}}^{2}\,ds\right]$
$\displaystyle\leq\,t\sup_{x\in\Gamma_{\delta}}\sum_{k\in\mathbb{N}}\left\|D\pi(x)e_{k}\right\|_{\mathbb{R}^{m}}^{2}\,<\,tM_{\pi}M_{B}.$

∎

######  Lemma 3.5.

Let Assumptions 1 & 2 hold. Then, $D\pi(x)Ly$, and hence
$D\pi(x)V(y)=D\pi(x)[Ly+N(y)]$, exists in $\mathbb{R}^{m}$ for all $x,y\in E$.
Moreover, $\partial_{t}\pi(\phi_{t}(x))=D\pi(\phi_{t}(x))V(\phi_{t}(x))$.

###### Proof.

Let $\\{e_{k}\\}_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ satisfying
Assumption 1, and write $y=\sum_{k\in E}y^{k}e_{k}$. Take $C_{\lambda}>0$ such
that (5) holds, so that uniformly in $k\in\mathbb{N}$

$\left(\lambda_{k+1}-\lambda_{k}\right)^{-1}\ln\frac{\lambda_{k+1}}{\lambda_{k}}\,\leq\,C_{\lambda}.$
(30)

Now, for $y\in E$ the sum

$Ly=\sum_{k\in\mathbb{N}}y^{k}\lambda_{k}e_{k}$

does not converge in $H$, but in some larger Hilbert space which we denote
$H^{L}$. Nevertheless, analogously to the definition of stochastic
convolutions with respect to white noise, we claim that

$D\pi(x)Ly\,=\,\sum_{k\in\mathbb{N}}y^{k}\lambda_{k}D\pi(x)e_{k}$

converges in $\mathbb{R}^{m}$. To prove this, recall that

$\pi(\phi_{t}(x))=\gamma^{-1}\left(\gamma_{\pi(x)}\right)\quad\text{ for all
$x\in\Gamma_{\delta}$ and $t>C_{\lambda}$.}$ (31)

Therefore, using the fact that $\alpha\mapsto\gamma_{\alpha}$ is a $C^{1}$ map
with bounded invertible first derivative and applying the chain rule to (31),
we have

$D\pi(x)e_{k}\,=\,M(x)D\phi_{t}(x)$

for a bounded linear operator $M(x)$ that is bounded uniformly in
$x\in\Gamma_{\delta}$ by some $c>0$. Then, using the mild formulation of
$D\phi_{t}$,

$\displaystyle\left\|D\pi(x)e_{k}\right\|_{\mathbb{R}^{m}}$
$\displaystyle\leq\,c\,\left\|D\phi_{t}(x)e_{k}\right\|_{E}$
$\displaystyle\leq\,c\,\left\|\Lambda_{t}e_{k}+\int_{0}^{t}\Lambda_{t-s}DN(\phi_{t}(x))D\phi_{t}(x)e_{k}\right\|_{E}$
$\displaystyle\leq\,c_{1}\,e^{-\lambda_{k}t}\left\|e_{k}\right\|_{E}+c_{2}\,\frac{1}{\lambda_{k}}\left(e^{-\lambda_{k}t}-1\right)\left\|e_{k}\right\|_{E}.$

It therefore follows that

$\left\|D\pi(x)Ly\right\|_{\mathbb{R}^{m}}\,\leq\,c_{1}\,\sum_{k\in\mathbb{N}}|y_{k}|\lambda_{k}e^{-\lambda_{k}t}+c_{2}\,\sum_{k\in\mathbb{N}}|y_{k}|+c_{2}\,\sum_{k\in\mathbb{N}}|y_{k}|e^{-\lambda_{k}t}.$
(32)

By choosing $t>C_{\lambda}$, the sequence
$\left(\lambda_{k}e^{-\lambda_{k}t}\right)_{k\in\mathbb{N}}$ is summable.
Since $\left(|y_{k}|\right)_{k\in\mathbb{N}}$ and
$\left(e^{-\lambda_{k}t}\right)_{k\in\mathbb{N}}$ are also summable, we see
that the quantities in (32) converge in $\mathbb{R}^{m}$. The statement
$\partial_{t}\pi(\phi_{t}(x))=D\pi(\phi_{t}(x))V(\phi_{t}(x))$ then follows
from the chain rule. ∎

######  Theorem 3.6.

Let Assumptions 1, 2, & 3 hold. Then, for any orthonormal basis
$\\{e_{k}\\}_{k\in\mathbb{N}}$ of $H$ contained in $E$,

$\displaystyle\pi(X_{t\wedge\tau})$
$\displaystyle=\,\pi(X_{0})+\int_{0}^{t\wedge\tau}D\pi(X_{s})V(X_{s})\,ds+\frac{\sigma^{2}}{2}\int_{0}^{t\wedge\tau}\sum_{k\in\mathbb{N}}D^{2}\pi(X_{s})[Be_{k},Be_{k}]\,ds$
(33) $\displaystyle\qquad+\sigma\int_{0}^{t\wedge\tau}D\pi(X_{s})B\,dW_{s},$

and all of the terms in the above expression are well-defined in
$\mathbb{R}^{m}$.

###### Proof.

Take $t>0$ and a partition of $[0,t]$ with boundary points
$\\{t_{k}\\}_{k=1}^{M}$. Let $t^{*}\coloneqq t\wedge\tau$ and
$t_{i}^{*}\coloneqq t_{i}\wedge\tau$ for $i\in\\{1,\ldots,M\\}$. By Theorem
3.1, $\pi(X_{t^{*}})$ may be written as

$\pi(X_{t^{*}})\,=\,\pi(X_{0})+\sum_{i=1}^{M}\Big{(}D\pi(X_{t^{*}_{i}})[X_{t^{*}_{i+1}}-X_{t^{*}_{i}}]+D^{2}\pi(w_{i})[X_{t^{*}_{i+1}}-X_{t^{*}_{i}},X_{t^{*}_{i+1}}-X_{t^{*}_{i}}]\Big{)},$
(34)

where $w_{i}=a_{i}X_{t^{*}_{i}}+(1-a_{i})X_{t^{*}_{i+1}}$ for some
$a_{i}\in[0,1]$. Since $(X_{t})_{t\geq 0}$ is a mild solution of (2),

$\displaystyle X_{t^{*}_{i+1}}-X_{t^{*}_{i}}$
$\displaystyle=\,\left[\Lambda_{t^{*}_{i+1}-t^{*}_{i}}-I\right]X_{0}+\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}N(X_{s})\,ds$
$\displaystyle\qquad+\sigma\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}B\,dW_{s}$
$\displaystyle\eqqcolon\,U^{1}_{i}+U^{2}_{i}+\sigma U_{i}^{3}.$

Inserting this into (34), we then have

$\displaystyle\pi(X_{t^{*}})-\pi(X_{0})$
$\displaystyle=\,\sum_{i=1}^{M}D\pi(X_{t^{*}_{i}})\left[U^{1}_{i}\right]+\sum_{i=1}^{M}D\pi(X_{t^{*}_{i}})\left[U^{2}_{i}\right]+\sigma\sum_{i=1}^{M}D\pi(X_{t^{*}_{i}})\left[U_{i}^{3}\right]$
(35)
$\displaystyle\quad+\sum_{i=1}^{M}D^{2}\pi(w_{i})\left[U^{1}_{i}+U^{2}_{i},U^{1}_{i}+U^{2}_{i}\right]+2\sigma\sum_{i=1}^{M}D^{2}\pi(w_{i})\left[U^{1}_{i}+U^{2}_{i},U_{i}^{3}\right]$
$\displaystyle\qquad+\sigma^{2}\sum_{i=1}^{M}D^{2}\pi(w_{i})\left[U^{3}_{i},U^{3}_{i}\right]$
$\displaystyle\eqqcolon I+II+III+IV+V+VI.$

We will prove that, as the mesh size of the partition $\\{t_{i}\\}_{i=1}^{M}$
tends to zero, we have

$\displaystyle(I+II)\,\rightarrow\,\int_{0}^{t^{*}}D\pi(X_{s})V(X_{s})\,ds,\qquad
III\,\rightarrow\,\int_{0}^{t^{*}}D\pi(X_{s})B\,dW_{s},$ $\displaystyle
IV\,\rightarrow\,0,\qquad V\,\rightarrow\,0,\qquad\qquad
VI\,\rightarrow\,\int_{0}^{t^{*}}\sum_{k\in\mathbb{N}}D^{2}\pi(X_{s})\left[Be_{k},Be_{k}\right]\,ds.$

We first handle the nonzero limits.

For a fixed partition $\\{t_{i}\\}_{i=1}^{M}$ with mesh size $h>0$, choose
arbitrary $i\in\\{1,\ldots,M\\}$ and let $(\hat{X}_{s})_{s\in[t_{i},t_{i+1}]}$
be defined by the deterministic flow started at $X_{t_{i}}$,

$\hat{X}_{t_{i}}\,=\,X_{t_{i}},\quad\hat{X}_{s}\,=\,\Lambda_{s-t_{i}}X_{t_{i}}+\int_{t_{i}}^{s}\Lambda_{t-r}N(\hat{X}_{r})\,dr\quad\text{
for }\quad s\in[t_{i},t_{i+1}].$

For the limit of $(I+II)$, we apply the second order Taylor’s theorem to
$\pi(\hat{X}_{t^{*}_{i+1}})$ centered at $\hat{X}_{t_{i}}$ to obtain

$D\pi(\hat{X}_{t^{*}_{i}})\left[\hat{X}_{t^{*}_{i+1}}-\hat{X}_{t^{*}_{i}}\right]+D^{2}\pi(U^{5}_{i})\left[\hat{X}_{t^{*}_{i+1}}-\hat{X}_{t^{*}_{i}},\hat{X}_{t^{*}_{i+1}}-\hat{X}_{t^{*}_{i}}\right]\,=\,\pi(\hat{X}_{t^{*}_{i+1}})-\pi(\hat{X}_{t^{*}_{i}}),$

where $U^{5}_{i}$ is defined for some $a_{i}^{5}\in[0,1]$ as

$U_{i}^{5}\,\coloneqq\,a_{i}^{5}\hat{X}_{t^{*}_{i}}+(1-a_{i}^{5})U^{4}_{i}.$

By Grönwall’s inequality and the Lipschitz property of $N$ (on
$\Gamma_{\delta}$), it can be seen that

$\left\|X_{s}-\hat{X}_{s}\right\|_{E}\,\sim\,O(h^{2})\quad\text{ for
}\,\,\,s\in[t_{i},t_{i+1}].$ (36)

It then follows that

$\displaystyle\left\|\hat{X}_{t^{*}_{i+1}}-\hat{X}_{t^{*}_{i}}-\left[(\Lambda_{t^{*}_{i+1}-t^{*}_{i}}-I)X_{t^{*}_{i}}+\int_{t_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}N(X_{s})\,ds\right]\right\|_{E}$
$\displaystyle\leq\,K_{0}h\sup_{s\in[t^{*}_{i},t^{*}_{i+1}]}\left\|\hat{X}_{s}-X_{s}\right\|_{E}$
$\displaystyle\leq\,K_{1}h^{2}$

for some constants $K_{0},K_{1}>0$. Using (10), for all $t\in(t_{i},t_{i+1})$
we have,

$\left\|\hat{X}_{t^{*}}-\hat{X}_{t^{*}_{i}}\right\|_{E}\,\sim\,o(h^{1/2})$
(37)

Hence, using Lemma 3.5 each summand in $(I+II)$ is estimated as

$\displaystyle
D\pi(X_{t^{*}_{i}})\left[(\Lambda_{t^{*}_{i+1}-t^{*}_{i}}-I)X_{t^{*}_{i}}+\int_{t_{i}}^{t_{i+1}}\Lambda_{t_{i+1}-s}N(X_{s})\,ds\right]$
$\displaystyle=\,-D^{2}\pi(U^{5}_{i})\left[\hat{X}_{t^{*}_{i+1}}-\hat{X}_{t^{*}_{i}},\hat{X}_{t^{*}_{i+1}}-\hat{X}_{t^{*}_{i}}\right]$
$\displaystyle\qquad\qquad+O(h^{2})+\pi(\hat{X}_{t^{*}_{i+1}})-\pi(\hat{X}_{t^{*}_{i}})$
$\displaystyle=\,o(h)+\pi(\hat{X}_{t^{*}_{i+1}})-\pi(\hat{X}_{t^{*}_{i}})$
$\displaystyle=\,o(h)+hD\pi(\hat{X}_{t^{*}_{i}})V(\hat{X}_{t^{*}_{i}}),$

where in the last line we have taken

$\pi(\hat{X}_{t^{*}_{i+1}})=\pi(\hat{X}_{t^{*}_{i}})+D\pi(\hat{X}_{t^{*}_{i}})V(\hat{X}_{t^{*}_{i}})h+O(h^{2}),$

and have absorbed the $O(h^{2})$ terms into the $o(h)$ term. Noting that
$\hat{X}_{t_{i}^{*}}=X_{t_{i}^{*}}$, summing over $\\{t_{i}^{*}\\}_{i=1}^{M}$,
and letting $h\rightarrow 0$ yields the desired convergence.

For the limit of $III$, note that a Taylor expansion of $\Lambda_{t_{i+1}-s}$
about $s=t_{i+1}$ yields

$\Lambda_{t_{i+1}-s}\,=\,I+L\Lambda_{u}(t_{i+1}-s)$ (38)

for some $u\in(s,t_{i+1})$. Hence, for arbitrary $x\in\Gamma_{\delta}$

$\displaystyle
D\pi(x)\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}B\,dW_{s}$
$\displaystyle=\,\int_{t_{i}^{*}}^{t_{i+1}^{*}}D\pi(x)\left(I+(t_{i+1}-s)L\Lambda_{u}\right)B\,dW_{s}$
$\displaystyle=\,D\pi(x)B[W_{t_{i+1}^{*}}-W_{t_{i}^{*}}]+\int_{t_{i}^{*}}^{t_{i+1}^{*}}(t_{i+1}-s)D\pi(x)L\Lambda_{u}B\,dW_{s}.$

By Lemma 3.5, $D\pi(x)L\Lambda_{u}B:H\rightarrow\mathbb{R}^{m}$ is a bounded
linear operator, and by the Riesz Representation theorem we may consider
$D\pi(x)L\Lambda_{u}B$ as an element of $H^{m}$. Hence, applying the
Burkholder-Davis-Gundy inequality for Hilbert space valued local martingales
[27], we have

$\displaystyle\operatorname{\mathbb{E}}_{x}\left[\left\|\int_{t_{i}^{*}}^{t_{i+1}^{*}}(t_{i+1}-s)D\pi(x)L\Lambda_{u}B\,dW_{s}\right\|^{2}_{\mathbb{R}^{m}}\right]$
$\displaystyle\leq\,h\operatorname{\mathbb{E}}_{x}\left[\int_{t_{i}^{*}}^{t_{i+1}^{*}}\left\|D\pi(x)L\Lambda_{u}B\right\|^{2}_{H^{m}}\,ds\right]$
$\displaystyle\leq\,h^{2}\left\|D\pi(x)L\right\|^{2}\left\|B\right\|^{2}e^{-2\omega
t_{i}^{*}}$

Therefore for small $h>0$ and any $i\in\mathbb{N}$,

$D\pi(X_{t_{i}^{*}})\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}B\,dW_{s}\,=\,D\pi(X_{t_{i}^{*}})B[W_{t^{*}_{i+1}}-W_{t^{*}_{i}}]+o\left(h^{2}\right),$

and observing that $K=\lfloor t/h\rfloor$, we have the following in
probability

$\displaystyle III$
$\displaystyle=\,\sum_{i=0}^{K}D\pi(X_{t^{*}_{i}})\left[\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}B\,dW_{s}\right]$
$\displaystyle=\,\sum_{i=0}^{K}D\pi(X_{t^{*}_{i}})B[W_{t^{*}_{i+1}}-W_{t^{*}_{i}}]+o\left(h^{1/2}\right)$
$\displaystyle\qquad\xrightarrow[h\rightarrow
0]{}\,\int_{0}^{t^{*}}D\pi(X_{s})B\,dW_{s}.$

To obtain $VI$, we write $(W_{t})_{t\geq 0}$ as in (28), and note that by
Itô’s isometry,

$D^{2}\pi(w_{i})\left[\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{i+1}-s}B\sum_{k\in\mathbb{N}}e_{k}d\beta_{s}^{k},\,\int_{t^{*}_{i}}^{t^{*}_{i+1}}\Lambda_{t^{*}_{k+1}-s}B\sum_{\ell\in\mathbb{N}}e_{\ell}d\beta_{s}^{\ell}\right]$

is equivalent in mean square to

$\int_{t^{*}_{k}}^{t^{*}_{k+1}}\sum_{k\in\mathbb{N}}D^{2}\pi(w_{k})\left[\Lambda_{t^{*}_{k+1}-s}Be_{k},\,\Lambda_{t^{*}_{k+1}-s}Be_{k}\right]\,ds.$

Hence, again using (38), we have

$\displaystyle\int_{t^{*}_{i}}^{t^{*}_{i+1}}\sum_{k\in\mathbb{N}}D^{2}\pi(w_{i})\left[\Lambda_{t^{*}_{i+1}-s}Be_{k},\,\Lambda_{t^{*}_{i+1}-s}Be_{k}\right]\,ds$
$\displaystyle\qquad=\,\int_{t^{*}_{i}}^{t^{*}_{i+1}}\sum_{k\in\mathbb{N}}\bigg{(}D^{2}\pi(w_{i})\left[Be_{k},Be_{k}\right]$
$\displaystyle\qquad\qquad+2D^{2}\pi(w_{i})\left[L\Lambda_{u}Be_{k},Be_{k}\right](t^{*}_{i+1}-s)$
$\displaystyle\qquad\qquad+D^{2}\pi(w_{i})\left[L\Lambda_{u}Be_{k},L\Lambda_{u}Be_{k}\right](t^{*}_{i+1}-s)^{2}\bigg{)}\,ds$
$\displaystyle\qquad=\,(t^{*}_{i+1}-t^{*}_{i})\sum_{k\in\mathbb{N}}D^{2}\pi(w_{i})\left[Be_{k},Be_{k}\right]+O(h^{2}).$

Summing over $k\in\\{1,\ldots K\\}$ and letting $h\rightarrow 0$, we have
convergence in probability of $VI$.

Finally, we remark that $IV$ and $V$ tend to zero as $h\rightarrow 0$ by (10).
∎

##  Appendix A Solution Theory for Stochastic Reaction-Diffusion Systems

For the sake of completeness, we here show that Assumption 1 and an additional
hypothesis imply Assumption 3.

######  Proposition A.1.

Let Assumptions 1 & 3 hold, and assume that
$Y_{t}\coloneqq\int_{0}^{t}\Lambda_{t-s}B\,dW_{s}$ is a well-defined
continuous $D(N)$-valued process. Then, we have the following: For each
$X_{0}\in H$ there exists a unique local in time mild solution
$(X_{t})_{t\in[0,\tau)}$ to (2),

$X_{t}\,=\,\Lambda_{t}X_{0}+\int_{0}^{t}\Lambda_{t-s}N(X_{s})\,ds+\sigma\int_{0}^{t}\Lambda_{t-s}B\,dW_{s},\qquad
t\in[0,\tau).$ (39)

###### Proof.

We sketch the proof of existence and uniqueness of mild solutions to (2). Note
that for bounded $F\subset E$, the operator $N_{F}\coloneqq N\rvert_{F}$ is
locally Lipschitz on $E$ with Lipschitz constant $\kappa=\kappa(F)>0$. Fix an
arbitrary $T>0$. For some $E$-valued Markov process $(X^{\prime}_{t})_{t\geq
0}$ define $V_{t}\coloneqq X^{\prime}_{t}-Y_{t}$. Note that
$(X^{\prime}_{t})_{t\geq 0}$ satisfies (39) for $t\in[0,T]$ if and only if

$V_{t}\,=\,\Lambda_{t}X_{0}+\int_{0}^{t}\Lambda_{t-s}N_{F}(V_{s}+Y_{s})\,ds\qquad\text{
for $t\in[0,T]$}.$ (40)

Let $C_{T}$ be the space of continuous paths from $[0,T]$ to $D(N)$, and
define $U:C_{T}\rightarrow C_{T}$ by

$U(v)(t)\,=\,\Lambda_{t}X_{0}+\int_{0}^{t}\Lambda_{t-s}N_{F}(v_{s}+Y_{s})\,ds,\qquad
v\in C_{T},\,\,t\in[0,T].$

Due to the local boundedness of $(\Lambda_{t})_{t\geq 0}$ and $N$ on $E$, the
local inversion theorem [10, Lemma 9.2] implies that there exists small $T>0$
such that (40) admits a unique solution $(V_{t})_{t\in[0,T]}$, and hence there
exists a unique local mild solution $(X^{\prime}_{t})_{t\in[0,T]}$ to (2).

If $N$ is globally Lipschitz with Lipschitz constant $\kappa>0$, we can extend
this solution globally in time. To do so, note that

$e^{-\omega
t}\left\|V_{t}\right\|_{D(N)}\,\leq\,\left\|X_{0}\right\|_{D(N)}+\int_{0}^{t}\kappa
e^{-\omega s}\left\|Y_{s}\right\|_{D(N)}\,ds+\int_{0}^{t}\kappa e^{-\omega
s}\left\|V_{s}\right\|_{D(N)}\,ds.$

We may then apply Grönwall’s inequality to conclude that $V_{t}$ cannot blowup
in finite time, and we may extend the solution to (40) to $t\in[0,\infty)$. ∎

######  Remark A.2.

The assumption that $(Y_{t})_{t\geq 0}$ is a $D(N)$-valued process fails when
$B=I$, $L=\Delta$, $N$, is a polynomial, and $d\geq 2$. In this case, to make
sense of solutions to (2) one must perform a renormalization procedure [9, 13,
14], as discussed for stochastic reaction-diffusion equations by Berglund &
Kuehn [4], and for many other classes of SPDEs elsewhere in the literature. It
is presently unclear as to how much of the following discussion holds for (2)
when it must be renormalized.

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