#  h
Search for an invisible scalar in $t\bar{t}$ final states
at the LHC

Duarte Azevedo${}^{1,2\,}$111E-mail<EMAIL_ADDRESS>, Rodrigo
Capucha${}^{3\,}$222E-mail<EMAIL_ADDRESS>, Pedro
Chaves${}^{4\,}$333E-mail<EMAIL_ADDRESS>,
João Bravo Martins${}^{3\,}$444E-mail<EMAIL_ADDRESS>,
António Onofre${}^{4\,}$555E-mail<EMAIL_ADDRESS>Rui
Santos${}^{3,5\,}$666E-mail<EMAIL_ADDRESS>
1Institute for Theoretical Physics, Karlsruhe Institute of Technology,
Wolfgang-Gaede-Str. 1, 76128 Karlsruhe, Germany
2Institute for Astroparticle Physics, Karlsruhe Institute of Technology,
Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
3Centro de Física Teórica e Computacional, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, Edifício C8 1749-016 Lisboa, Portugal
4 Centro de Física da Universidade do Minho e Universidade do Porto (CF-UM-
UP),
Universidade do Minho, 4710-057 Braga, Portugal
5ISEL - Instituto Superior de Engenharia de Lisboa,
Instituto Politécnico de Lisboa 1959-007 Lisboa, Portugal

###### Abstract

We use the current $t\bar{t}$ experimental analysis to look for Dark Matter
(DM) particles hidden in the final state. We present a phenomenological study
where we successfully perform the reconstruction of a $t\bar{t}$ system in the
presence of a scalar mediator $Y_{0}$, that couples to both Standard Model
(SM) and to DM particles. We use a MadGraph5_aMC@NLO simplified DM model,
where signal samples of $pp\rightarrow t\bar{t}Y_{0}$ are generated at the
Large Hadron Collider (LHC) with both Charge-Parity (CP) -even and CP-odd
couplings of $Y_{0}$ to the top quarks. Different mass scales for the $Y_{0}$
mediator are considered, from the low mass region ($\sim$ 0 GeV) to masses
close to the Higgs boson mass (125 GeV). The dileptonic final states of the
$t\bar{t}$ system were used in our analysis. The reconstruction of the
$t\bar{t}$ system is done with a kinematic fit, without reconstructing the
mediator. All relevant SM backgrounds for the dileptonic $t\bar{t}$ search at
the LHC are considered. Furthermore, CP angular observables were used to probe
the CP-nature of the coupling between the mediator and top-quarks, which
allowed to set confidence level (CL) limits for those Yukawa couplings as a
function of the mediator mass.

##  1 Introduction

Evidence for the existence of dark matter (DM) provided by astrophysical
observations, be it from gravitational lensing effects [1, 2, 3, 4], galactic
rotational velocity curves [5] or from other measurements like the collision
of clusters of galaxies (Bullet Cluster) [6], is by now overwhelming. However,
the nature of this non-luminous matter remains unknown in spite of decades of
intensive searches in a vast array of experiments. From the observations, we
can hypothesise that if DM is actually composed of particles, they only
interact very weakly with the Standard Model (SM). Since SM neutrinos have
been mostly ruled out as possible DM candidates, due to cosmological
constraints [7], this suggests the possible existence of a ”dark sector”
composed of particles beyond the SM. Attempts at direct and indirect detection
of the myriad of particles proposed from extending the SM with such a sector
(such as weakly interacting massive particles and axions, among many others)
have, so far, provided no definitive results favouring any particular
hypotheses [8, 9].

DM searches at colliders have focused on mono-events, such as mono-jet, mono-
Higgs among others [10, 11, 12]. There are also bounds on the portal couplings
from the invisible branching ratio of the 125 GeV Higgs [13, 14]. Searches for
DM particles produced alongside a $t\bar{t}$ pair and single top quark events
have also been performed in the past [15, 16, 17, 18] mostly using variables
such as missing energy as a discriminator, with no attempt to reconstruct the
kinematics of the top quarks. However, and to best of our knowledge, this is
the first time a study is designed to look for DM particles hidden in the
current on-going searches and analysis. Let us consider as an example the case
of $t\bar{t}$ production in the di-leptonic channel. The question we want to
answer is: if a very light DM particle would be produced alongside the
$t\bar{t}$ pair would we see any difference in any distributions? If no
differences are found, could the analysis already performed be used to limit
the couplings of DM particles for very light invisible particles? One should
note that from the point of view of high energy collider physics we are really
exploring the case of a zero DM mass particle which means that we are spanning
DM masses from the scenarios of ultra-light particles to a few GeV in mass.
That is, for LHC energies, masses below about 1 GeV are for all practical
purposes equal to zero. Still we will also show how the $t\bar{t}$ analysis
performs in the case of a 125 GeV scalar.

In order to implement our idea we consider a new mediator that couples to both
DM and to the SM particles. We study the possibility of using previous
mainstream experimental analysis of $pp\to t\bar{t}$ in the di-leptonic
channel to gauge the impact of a spin-0 DM mediator ($J^{CP}=0^{\pm}$) in the
associated production process $t\bar{t}Y_{0}$. The analysis is performed
within the description of simplified models of DM production at the LHC, where
the DM mediator ($Y_{0}$) couples to the top-quarks proportionally to the top
mass. The results are presented as a function of the modifier of this new
Yukawa coupling. This is a convenient approach, as the LHC can explore a large
spectrum of DM mediator masses and coupling strengths, allowing to access the
CP-nature of these mediators, in case they exist even as mixed CP states.

This paper is organised as follows: the simplified DM model, the relevant
parameters and the angular observables we used, are presented in Section 2.
The event generation and simulation are described in Section 3 and, in Section
4, the event selection and kinematic reconstruction are discussed. Our results
are presented in Section 5 and the main conclusions are described in Section
6.

##  2 The DM Lagrangian

In our study, we used the simplified DM model DMsimp [19] where, besides the
scalar $Y_{0}$ boson, we also have a dark sector that couples only to $Y_{0}$
. In our paper, we will remain agnostic to the latter, focusing solely on the
interaction between the $Y_{0}$ DM mediator to the SM content. In particular,
we will assume Yukawa couplings proportional to the mass of the respective SM
particle and hence dedicate ourselves exclusively to top quarks. The
Lagrangian density can thus be simplified and written as follows

$\mathcal{L}_{SM}^{Y_{0}}=\frac{y_{33}^{t}}{\sqrt{2}}\bar{t}(g_{u_{33}}^{S}+ig_{u_{33}}^{P}\gamma^{5})tY_{0}\quad,$
(2.1)

where the $g^{S/P}_{u_{33}}$ are the CP-even/-odd couplings of the DM mediator
($Y_{0}$) to top quarks, respectively. They are normalized to the SM Yukawa
couplings, $y^{f}_{ii}=\sqrt{2}m_{f}/v$. The scalar hypothesis (CP=+1) is
given by setting $g^{S}_{u_{33}}=1$ and $g^{P}_{u_{33}}=0$ and for the pseudo-
scalar scenario (CP=-1) we set $g^{S}_{u_{33}}=0$ and $g^{P}_{u_{33}}=1$. When
both $g^{S/P}_{u_{33}}\neq 0$, the interaction has both CP-even and -odd
components and is thus CP-violating.

Our goal is to explore how angular distributions of SM particles may help
probing the dark sector, by looking into the expected changes of these
observables in the presence of this DM mediator. Several CP-observables have
been proposed in the literature to probe the CP-nature of the coupling of top
quarks to the Higgs boson at the LHC or future colliders, using mainly the
$t\bar{t}H$ channel [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. The vast
majority of these variables are only sensitive to the square terms
$(g^{S}_{u_{33}})^{2}$ and $(g^{P}_{u_{33}})^{2}$ that appear in the cross
section of the interaction described by equation 2.1.

After looking in detail at several possible observables, we concluded that the
most effective ones are the azimuthal angle difference
$\Delta\phi_{\ell^{-}\ell^{+}}$ of the charged leptons that come from the
decay of top quarks, and the $b_{4}$ variable evaluated in the laboratory
frame (LAB),
$b_{4}=(p^{z}_{t}.p^{z}_{\bar{t}})/(|\vec{p}_{t}|.|\vec{p}_{\bar{t}}|)$, where
the $z$-direction corresponds to the beam direction, and
$\vec{p}_{t(\bar{t})}$ and $p^{z}_{t(\bar{t})}$ correspond to the total and
z-component of the top (anti-top) quark momentum measured in the LAB frame,
respectively. It is worth noting that the $b_{4}$ variable depends on the $t$
and $\bar{t}$ polar angles, $\theta_{t}$ and $\theta_{\bar{t}}$ respectively,
with respect to the $z$-direction, and can be expressed as
$b_{4}=\cos{\theta_{t}}\times\cos{\theta_{\bar{t}}}$. In order to evaluate
this variable, the kinematic reconstruction of the $t\bar{t}$ system needs to
be accomplished. This will be discussed in the next sections.

##  3 Event generation and simulation

LHC-like signal and background events were generated with MadGraph5_aMC@NLO
[51], with a centre-of-mass energy of 13 TeV. The DM simplified model, DMsimp
[19]777Available in the FeynRules repository., was used to generate $pp\to
t\bar{t}Y_{0}$ signal events, at Leading Order (LO). The pure scalar and
pseudo-scalar signals were generated by setting the respective couplings as
mentioned in the previous section (following the Lagrangian density in
equation 2.1). The mass of the DM mediator was set to $m_{Y_{0}}=0,1,10$ and
$125$ GeV, and the top quark mass to $m_{t}=172.5~{}GeV$. Only the dileptonic
final state of the $t\bar{t}$ system was considered ($t\bar{t}\rightarrow
bW^{+}\bar{b}W^{-}\rightarrow
b\ell^{+}\nu_{\ell}\bar{b}\ell^{-}\bar{\nu}_{\ell}$), with the DM mediator
forced not to decay, although if we allow the DM mediator to decay to mostly
DM particles, the analysis and subsequent results do not change. Backgrounds
from SM $t\bar{t}$ (plus up to 3 jets), $t\bar{t}V$ (plus up to 1 jet),
$t\bar{t}H$, single top quark production ($t$-, $s$\- and $Wt$-channels),
$W/Z$ (plus up to 4 jets), $W$($Z$)$b\bar{b}$ (plus up to 2 jets) and
$WW,ZZ,WZ$ diboson processes were also generated using MadGraph5_aMC@NLO at
LO. Following event generation and hadronization by PYTHIA [52], all signal
and background events went through a fast simulation of a typical LHC detector
performed by Delphes [53], using the default ATLAS detector cards. Further
details on the event generation and detector simulation can be found in [44].
The analysis of signal and background events is performed within the
MadAnalysis5 [54] framework.

##  4 Event selection and kinematic reconstruction

Events are selected by requiring the jets and leptons reconstructed by Delphes
to have their pseudo-rapidity888The pseudo-rapidity is defined by
$\eta=-ln[tan(\theta/2)]$, where $\theta$ is the particle polar angle.
$\eta<2.5$ and transverse momenta $p_{T}>20$ GeV. Only events with two jets
and two isolated leptons of opposite charge are accepted. To avoid
contamination from the $Z$ \+ jets background, we require the invariant mass
of the two lepton system to fulfil $|m_{\ell^{+}\ell^{-}}-m_{Z}|>10$ GeV.
Further details on the event selection criteria can be found in [45].

In the reconstruction of the $t\bar{t}$ system, we assume the reconstructed
leptons are the ones from the W decays, originated in the top quark decays. We
then need to assign the reconstructed jets to the correct b-quarks from the
associated top quarks. In order to avoid mismatches, once two jets are present
in the final state, we use multivariate statistical methods from the Toolkit
for Multivariate Data Analysis, TMVA [55], to find the right pairing of jets
and leptons..

Figure 1: Normalized TMVA input variable distributions for correct
combinations (labeled as signal, in blue) and wrong combinations (labeled as
background, in red), for a DM $J^{P}=0^{+}$ mediator. The $\Delta R$ between
the $\ell^{+}$ and the $b$-jet from the $t$ decay, is shown on the left plot.
The corresponding $\Delta\Phi$ distribution can be seen on the right.

Figure 2: Normalized TMVA input variable distributions for correct
combinations (labeled as signal, in blue) and wrong combinations (labeled as
background, in red), for a DM $J^{P}=0^{+}$ mediator. The $\Delta\theta$
between the $\ell^{+}$ and the $b$-jet from the $t$ decay is shown on the left
and the lepton plus $b$-jet mass difference is shown on the right.

To that effect we use several distributions (which already include
hadronization and shower effects), where right and wrong combinations are
compared. A wrong combination happens whenever a jet not originating from a
top decay is assigned to its corresponding b-quark. Right (labelled as Signal,
in blue) and wrong (labelled as Background, in red) combinations are
represented in Figures 1 and 2, for the $\Delta R$ 999$\Delta
R\equiv\sqrt{\Delta\Phi^{2}+\Delta\eta^{2}}$, where $\Delta\Phi\,(\Delta\eta)$
correspond to the difference in the azimuthal angle (pseudo-rapidity) of two
objects., $\Delta\Phi$ and $\Delta\theta$ between the $\ell^{+}$ lepton and
the jet from the hadronization of the $b$-quark (originated in the $t$ decay
and labelled as $b_{t}$). Similar distributions were also obtained for the
$\ell^{-}$ lepton, and used to optimize finding the right combination. Clear
differences between the wrong and right combinations are visible in all
distributions shown. The invariant mass difference between the pairs
($b_{t},l^{+}$) and ($\bar{b}_{\bar{t}},l^{-}$) is also shown in Figure 2, for
the right and wrong combinations. Several multivariate statistical methods
were then trained by TMVA, using right and wrong combinations distributions
for training and testing. All individual distributions were combined into a
single discriminant classifier for each one of the methods. In Figure 3
(left), we show the Receiver Operating Characteristic (ROC) curve for the
$Y_{0}$ scalar ($J^{P}=0^{+}$) and pseudo-scalar ($J^{P}=0^{-}$) mediators in
the top and bottom plots, respectively. From the ROC curves we can see that
the best method is a Boosted Decision Tree with Gradient boost (BDTG). The
corresponding classifier outputs are shown in Figure 3 (right), for scalar and
pseudo-scalar DM mediators in the top and bottom plots, respectively. From the
comparison between both cases, we can see that the scalar mediator is more
challenging, with a slightly worse ROC curve. For this reason, from this point
on, all results shown originate directly from the scalar mediator analysis,
since this case represents the most conservative scenario.

Figure 3: Background rejection versus signal acceptance (ROC curve) for
different multi-variate methods are compared, for the $J^{P}=0^{+}$ mediator
(top left). The distribution of the best classifier (BDTG) is also shown (top
right). The equivalent plots for the case of the $J^{P}=0^{-}$ mediator, are
shown in the bottom plots.

To reconstruct the 3-momentum of the undetected neutrinos from the top quark
decays, we impose the following energy-momentum conservation conditions to
events,

$\displaystyle(p$ ${}_{\nu}+p_{\ell^{+}})^{2}=m^{2}_{W},$ $\displaystyle(p$
${}_{\bar{\nu}}+p_{\ell^{-}})^{2}=m^{2}_{W},$ $\displaystyle(p$
${}_{W^{+}}+p_{b})^{2}=m^{2}_{t},$ (4.1) $\displaystyle(p$
${}_{W^{-}}+p_{\bar{b}})^{2}=m^{2}_{\bar{t}},$ $\displaystyle p$
${}^{x}_{\nu}+p^{x}_{\bar{\nu}}=\not{E}^{x},$ $\displaystyle p$
${}^{y}_{\nu}+p^{y}_{\bar{\nu}}=\not{E}^{y},$

where $p_{\varsigma}$ ($p_{\varsigma}^{\kappa}$) represents the four-momentum
of particle $\varsigma$ (its projection along the $\kappa$-axis). In the first
four equations mass constraints are imposed, where neutrinos and charged
leptons are assumed to reconstruct the masses of the $W$ bosons they
originated from which, when combined with the right jet, should reconstruct
the correspondent top quark masses. We also consider (last two equations) the
total missing transverse energy is wholly accounted for by the neutrinos. In
this approximation, we are assuming the DM mediator contribution to the
missing transverse energy to be negligible (as well as its $z$-axis component)
when compared to the neutrinos contribution. In order to find the best
solution for each event, the top quark and $W$ boson mass values, used by the
fit, were sampled 500 times from 2-dimensional probability distribution
functions (p.d.f.s) obtained from parton level (with shower effects)
$t\bar{t}Y_{0}$ signal events (see [45], for more details).

Figure 4: Two-Dimensional distributions of $t\bar{t}Y_{0^{+}}$ events:
generator-level transverse momentum versus reconstructed transverse momentum
for several particles (neutrino, top left; $t$, top right; $t\overline{t}$
system, bottom left; W+ boson, bottom right), for the massless DM mediator.

The neutrinos four-momenta are obtained from solving the set of equations 4.1,
above. If a solution is found, new mass values are tried around the found
value (up to 6 times), to check if neighbour mass points can indeed provide a
better fit overall. Due to the quadratic form of the mass equations, multiple
solutions may exist for a single event. We chose the solution that provides
the highest value of the likelihood ($L$) constructed using parton level (with
shower effects) distributions, in particular the p.d.f.s for the transverse
momenta of the neutrinos, top quarks and $t\bar{t}$ system, $P(p_{T_{\nu}})$,
$P(p_{T_{\bar{\nu}}})$, $P(p_{T_{t}})$, $P(p_{T_{\bar{t}}})$ and
$P(p_{T_{t\bar{t}}})$, respectively. This likelihood is defined according to

$L\propto\frac{1}{p_{T_{\nu}}p_{T_{\bar{\nu}}}}P(p_{T_{\nu}})P(p_{T_{\bar{\nu}}})P(p_{T_{t}})P(p_{T_{\bar{t}}})P(p_{T_{t\bar{t}}}),$
(4.2)

where the normalization factor ${1}/{p_{T_{\nu}}p_{T_{\bar{\nu}}}}$ is applied
to account for the energy losses due to radiation emission and effects from
detector resolutions which will increase the reconstructed neutrino
4-momentum. This factor compensates for values of neutrino $p_{T}$ which are
too extreme by giving less weight to those solutions. If no solution is found,
the event is discarded. For a scalar (pseudo-scalar) DM mediator mass of 0
GeV, we found that 75% (72%) of all events were correctly reconstructed, a
number that matches quite well typical SM $t\bar{t}$ analyses where such
kinematic reconstruction is attempted.

In Figure 4, we show the 2-dimensional $p_{T}$ distributions of the neutrino
(top left), top quark (top right), $t\bar{t}$ system (bottom left) and $W^{+}$
boson (bottom right), after kinematic reconstruction. We can see that the
parton level (with shower effects) and the reconstructed kinematics are highly
correlated for all particles or systems of particles. This implies that, after
experimental effects are taken into account, it is still possible to
reconstruct the $t\bar{t}$ system without even trying to reconstruct the
invisible DM mediator. This point is quite important since it opens up the
possibility of studying the changes in angular distributions of $t\bar{t}$
systems in the presence of a new invisible particle.

##  5 Results and Discussion

In Figure 5, we show the $b_{4}$ (left) and $\Delta\phi_{\ell^{+}\ell^{-}}$
(right) distributions after event selection and kinematic reconstruction, for
a reference luminosity of 100 fb-1. All SM backgrounds: $t\bar{t}$
($t\bar{t}c\bar{c}$ and $t\bar{t}$+light jets), $t\bar{t}b\bar{b}$,
$t\bar{t}V$, $t\bar{t}H$, single top quark production ($t$-, $s$\- and
$Wt$-channels), $W/Z$+jets, and diboson ($WW,ZZ,WZ$) events are represented.
The $t\bar{t}Y_{0}$ scalar and pseudo-scalar signals, with $m_{Y_{0}}=0$ GeV,
are shown as well, scaled by factors of 2 and 500 respectively, for
convenience. As expected, the main SM background contribution is the
$t\bar{t}$ due to its similarity with the signal final state topology. All
other backgrounds are essentially residual to the overall SM background
contribution. Differences in the shapes of the background distributions can
also be noticed when compared with the signals. For instance, in the $b_{4}$
distribution and for the scalar signal (in brown), events tend to populate
positive values more than negative values. This behaviour is inverted for the
pseudo-scalar case (in orange). For the $\Delta\phi_{\ell^{+}\ell^{-}}$
distribution (which is symmetric around zero), the scalar and pseudo-scalar
signals populate differently its extreme regions. For completeness, we show in
Figure 6 the missing transverse energy ($E_{T}$) distribution, which shows a
quite similar behaviour to the SM background one, for both the CP-even and CP-
odd signals. This means that missing $E_{T}$ is not a good discriminating
variable for this process which is a very important point to make in an
analysis for a process with DM in the final state.

Figure 5: The $b_{4}$ (left) and $\Delta\phi_{\ell^{+}\ell^{-}}$ (right)
distributions for scalar and pseudo-scalar signals (dashed curves) together
with the SM processes (full lines) with dileptonic final states, are
represented after event selection and kinematic reconstruction (exp), for a
reference luminosity of 100 fb-1. Scaling factors are applied to the scalar
and pseudo-scalar signals for convenience. Figure 6: Missing transverse energy
($E_{T}$) distributions for scalar and pseudo-scalar signals (dashed curves)
together with the SM processes (full lines) with dileptonic final states, are
represented after event selection and kinematic reconstruction (exp), for a
reference luminosity of 100 fb-1. Scaling factors are applied to the scalar
and pseudo-scalar signals for convenience.

The $b_{4}$ and $\Delta\phi_{\ell^{+}\ell^{-}}$ distributions were then used
to set confidence level limits (CLs) on the exclusion of the SM with a new CP-
mixed massless DM mediator particle, $Y_{0}$, assuming the SM hypothesis as
the null hypothesis (Scenario 1). The result is shown in Figure 7, for an
integrated luminosity corresponding roughly to the RUN 2 luminosity plus the
contribution from the first year of RUN 3, i.e., $L\sim 200$ fb-1. The CL
limits are shown as contour plots in the $(g^{S}_{u_{33}},g^{P}_{u_{33}})$ 2D
plane. It is clear that the CLs are identical for both observables, in this
scenario.

Figure 7: CLs for the exclusion of the SM with a massless DM mediator,
$Y_{0}$, with mixed scalar and pseudo-scalar couplings with the top quarks,
against the SM as null hypothesis, for the $\Delta\phi$ between the charged
leptons, $\Delta\phi_{\ell^{+}\ell^{-}}$ (left), and $b_{4}$ (right)
observables. Limits are shown for a luminosity of $L=200$ fb-1.

The CLs are also evaluated, for Scenario 1, for the full luminosity expected
at the end of the High-Luminosisty phase of the LHC (HL-LHC), for $L=3000$
fb-1, using the $\Delta\phi_{\ell^{+}\ell^{-}}$ distribution. The resulting
68% and 95% exclusion limits, for both luminosity values, are in Table 1. For
$L=3000$ fb-1, we observe a substantial improvement by factors of 2 to 3, on
the exclusion limits. Quite similar results where obtained when using a simple
counting experiment. This leads us to conclude that the observable choice has
little to no impact on the exclusion limits in this scenario and the DM
mediator production cross section is, in itself, the dominant factor.

Exclusion Limits |  $L$ = 200 fb-1 |  $L$ = 3000 fb-1
---|---|---
from $\Delta\phi_{l^{+}l^{-}}$ | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$m_{Y_{0}}$ = 0 GeV | $g^{S}_{u_{33}}\in$ | [-0.067, +0.067] | [-0.125, +0.125] | [-0.022, +0.022] | [-0.052, +0.052]
$g^{P}_{u_{33}}\in$ | [-0.91, +0.91] | [-1.71, +1.71] | [-0.44, +0.44] | [-0.85, +0.85]
Table 1: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings for fixed
luminosities of 200 fb-1 and 3000 fb-1 of the SM plus $Y_{0}$, assuming the SM
as the null hypothesis. The limits are shown at confidence levels of 68% and
95%, for the $\Delta\phi_{l^{+}l^{-}}$ variable.

Figure 8: CLs for the exclusion of the SM with a massless DM mediator,
$Y_{0}$, with mixed scalar and pseudo-scalar couplings with the top quarks,
against the SM plus a pure scalar DM mediator, for the $\Delta\phi$ between
the charged leptons (left). For completeness, the results for a simple event
counting experiment (one Bin) is also shown (right). Limits are represented
for a luminosity of $L=200$ fb-1.

For completeness, an alternative scenario was considered (Scenario 2), where
we assumed as null hypothesis the SM plus a pure CP-even DM mediator of mass 0
GeV. The main goal of this scenario is to quantify how well could the mixed
state be excluded from a pure CP-even mediator, in case of a discovery. This
scenario was explored by using the $\Delta\phi_{\ell^{+}\ell^{-}}$
distribution as well as the simple counting experiment used above for Scenario
1. The results are shown in Figure 8. Here, however, the difference between
both distributions is quite clear, i.e., the 68% CLs are much worse in the
latter case. This indicates that, in Scenario 2, in contrast with Scenario 1,
the chosen observable will have an important impact on the exclusion limits.
This also means that angular observables can indeed help on studying the CP-
nature of DM mediators upon discovery.

Lastly, to extend our results to a massive $Y_{0}$ produced together with
pairs of top quarks, additional signal events were generated with mediator
masses ($m_{Y_{0}}$) set to 1, 10 and 125 GeV. The selection criteria and
reconstruction procedure described in Section 4 were the same. The resulting
$\Delta\phi_{\ell^{+}\ell^{-}}$ distributions were then used to set CLs, in
both Scenarios 1 and 2, for a luminosity of 200 fb-1 and 3000 fb-1 as before.
The exclusion limits are depicted for all masses in Figure 9, for $L=3000$
fb-1. The respective 68% and 95% exclusion limits for Scenario 1 are shown in
Table 2, for both $L=200$ fb-1 and $L=3000$ fb-1. As expected, exclusion
limits worsen as masses increase in both Scenarios, since the $t\bar{t}Y_{0}$
production cross section decreases for heavier $Y_{0}$ masses, and improve
with increasing luminosity values. Also, notice that the observable choice
having a very small impact on the exclusion limits in Scenario 1 is true only
for smaller DM mediator masses, where the $Y_{0}$ production cross section is
the dominant factor contributing to the exclusion limits.

Figure 9: CLs for the exclusion of the SM with a massive DM mediator, $Y_{0}$ ($m_{Y_{0}}=$ 1, 10 and 125 GeV in the top, middle, and bottom rows, respectively), with mixed scalar and pseudo-scalar couplings, against the SM as null hypothesis (left), for the $\Delta\phi$ between the charged leptons. For completeness, the corresponding CL for the exclusion of the SM plus a mixed DM mediator against the SM plus a pure scalar DM mediator is also shown for $\Delta\phi$ (right). Limits are shown for a luminosity corresponding to the full HL-LHC luminosity ($L=3000$ fb-1). Exclusion Limits |  $L$ = 200 fb-1 |  $L$ = 3000 fb-1
---|---|---
from $\Delta\phi_{l^{+}l^{-}}$ | (68% CL) | (95% CL) | (68% CL) | (95% CL)
$m_{Y_{0}}$ = 1 GeV | $g^{S}_{u_{33}}\in$ | [-0.073, +0.073] | [-0.142, +0.142] | [-0.038, +0.038] | [-0.068, +0.068]
$g^{P}_{u_{33}}\in$ | [-0.89, +0.89] | [-1.65, +1.65] | [-0.43, +0.43] | [-0.83, +0.83]
$m_{Y_{0}}$ = 10 GeV | $g^{S}_{u_{33}}\in$ | [-0.198, +0.198] | [-0.368, +0.372] | [-0.098, +0.098] | [-0.188, +0.188]
$g^{P}_{u_{33}}\in$ | [-0.87, +0.87] | [-1.65, +1.65] | [-0.44, +0.44] | [-0.83, +0.83]
$m_{Y_{0}}$ = 125 GeV | $g^{S}_{u_{33}}\in$ | [-0.328,+0.322] | [-0.608, +0.612] | [-0.162, +0.162] | [-0.308, +0.308]
$g^{P}_{u_{33}}\in$ | [-1.48, +1.49] | [-2.77, +2.78] | [-0.75, +0.75] | [-1.41, +1.41]
Table 2: Exclusion limits for the $t\bar{t}Y_{0}$ CP-couplings for fixed
luminosities of 200 fb-1 and 3000 fb-1 of the SM plus $Y_{0}$, assuming the SM
as null hypothesis, for $Y_{0}$ masses of 1 GeV (top), 10 GeV (middle) and 125
GeV (bottom). The limits are shown at confidence levels of 68% and 95%, for
the $\Delta\phi_{l^{+}l^{-}}$ variable.

##  6 Conclusions

In this work we have explored the idea of using on-going searches and
measurements, such as the analysis that leads to the measurement of the
$t\bar{t}$ production cross section at the LHC, to look for hidden DM
particles in the final states. To that end, we present a new approach to fully
reconstruct the kinematics of the $t\bar{t}$ system present in $t\bar{t}Y_{0}$
events produced at the LHC. Our study was performed within the context of
simplified models of DM production at colliders. In this kinematic
reconstruction, the missing transverse energy was fully attributed to the
undetected neutrinos and no attempt to reconstruct the invisible DM mediator
was tried. The approximations used in our work appear to be valid in a wider
range of the DM mediator mass (starting at $m_{Y_{0}}=0$ GeV), according to
the resulting correlations between the generated and reconstructed kinematics.
An example of these correlations is shown in Figure 4 for the $m_{Y_{0}}=0$
GeV case. Moreover, we have checked that the pairing of the $b$-jets and
charged leptons originated from the same parent top quark decay was very well
achieved using several angular distributions and dedicated multivariate
statistical methods.

We have analyzed a significant number of angular observables, from which two
of them were selected to illustrate our findings, the
$\Delta\phi_{\ell^{+}\ell^{-}}$ and $b_{4}$ distributions. These observables
were shown to be sensitive not only to DM mediators with different mass
scales, but also with different CP-natures, in what concerns their couplings
to heavy SM particles. These distributions were then used to set exclusion
limits assuming the SM as the null hypothesis, for a luminosity of 200 fb-1
and 3000 fb-1, corresponding to the full luminosity of the High-Luminosity
Phase of the LHC (HL-LHC). We also considered a benchmark scenario that takes
into account, as null hypothesis, the SM plus a pure scalar DM mediator, in
order to check how sensitive the analysis is to a possible CP-mixed nature of
the DM mediator. We observe that, in the former case, the 95% CL limits using
the $\Delta\phi_{\ell^{+}\ell^{-}}$ angular distribution were
$g^{S}_{u_{33}}\in[-0.125,0.125]$ and $g^{P}_{u_{33}}\in[-1.71,1.71]$ for a
luminosity of 200 fb-1, and $g^{S}_{u_{33}}\in[-0.052,0.052]$ and
$g^{P}_{u_{33}}\in[-0.85,0.85]$ for a luminosity of 3000 fb-1. We have also
checked that a simple counting experiment can provide similar exclusion
limits. In the second case, we observed that the use of angular distributions
can improve the exclusion limits for the pseudo-scalar coupling by at least a
factor of two, if we want to understand the CP-nature of the DM mediator
couplings to SM particles. Finally, we extended our study to the case of a
massive DM mediator with $m_{Y_{0}}$ set to 1, 10 and 125 GeV. We observed
that the exclusion limits set from the $\Delta\phi_{\ell^{+}\ell^{-}}$
distributions got worse for increasing $Y_{0}$ masses, since the
$t\bar{t}Y_{0}$ cross section decreases in that case.

### Acknowledgments

R.C. and R.S. are partially supported by the Portuguese Foundation for Science
and Technology (FCT) under Contracts no. UIDB/00618/2020, UIDP/00618/2020,
CERN/FIS-PAR/0025/2021 CERN/FIS-PAR/0010/2021 and CERN/FIS-PAR/0021/2021. R.C.
is additionally supported by FCT Grant No. 2020.08221.BD. A.O. is partially
supported by FCT, under the Contract CERN/FIS-PAR/0037/2021. D.A. is supported
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
grant 396021762 - TRR 257.

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# Large deviations principle via Malliavin calculus for the Navier–Stokes
system driven by a degenerate white-in-time noise

Vahagn Nersesyan 111NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai,
3663 Zhongshan Road North, Shanghai, 200062, China, e-mail:
<EMAIL_ADDRESS>222Université Paris-Saclay, UVSQ, CNRS, Laboratoire
de Mathématiques de Versailles, 78000, Versailles, France Xuhui Peng 333MOE-
LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha,
Hunan 410081, China; e-mail<EMAIL_ADDRESS>Lihu Xu 444Department of
Mathematics, Faculty of Science and Technology, University of Macau, Macau,
China; e-mail<EMAIL_ADDRESS>
###### Abstract

The purpose of this paper is to establish the Donsker–Varadhan type large
deviations principle (LDP) for the two-dimensional stochastic Navier–Stokes
system. The main novelty is that the noise is assumed to be highly degenerate
in the Fourier space. The proof is carried out by using a criterion for the
LDP developed in [JNPS18] in a discrete-time setting and extended in [MN18] to
the continuous-time. One of the main conditions of that criterion is the
uniform Feller property for the Feynman–Kac semigroup, which we verify by
using Malliavin calculus.

AMS subject classification: 35Q30, 37H15, 60B12, 60F10, 60H07, 93B05

Keywords: Navier–Stokes system, large deviations, Malliavin calculus,
degenerate noise, Feynman–Kac semigroup, approximate controllability,
multiplicative ergodicity, uniform Feller property

###### Contents

  1. 0 Introduction
  2. 1 Main results
    1. 1.1 LDP and multiplicative ergodicity
    2. 1.2 Proof of Theorem 1.2
  3. 2 Elements of Malliavin calculus
  4. 3 Proof of uniform Feller property
    1. 3.1 Reduction to a gradient estimate
    2. 3.2 Proof of Proposition 3.2

##  0 Introduction

In this paper, we study the large deviations principle (LDP) for the
incompressible Navier–Stokes (NS) system on the torus
${\mathbb{T}}^{2}={\mathbb{R}}^{2}/2\pi{\mathbb{Z}}^{2}$:

$\partial_{t}u-\nu\Delta u+\langle u,\nabla\rangle u+\nabla
p=\eta(t,x),\quad\mathop{\rm div}\nolimits u=0,\quad x\in{\mathbb{T}}^{2}.$
(0.1)

Here $u=(u_{1}(t,x),u_{2}(t,x))$ and $p=p(t,x)$ are the unknown velocity field
and pressure of the fluid, $\nu>0$ is the viscosity, and $\eta$ is an external
random force. We consider this system in the usual space

$H=\Bigg{\\{}u\in
L^{2}({\mathbb{T}}^{2},{\mathbb{R}}^{2}):\,\int_{{\mathbb{T}}^{2}}u(x){\textup{d}}x=0,\,\,\,\mathop{\rm
div}\nolimits u=0\,\,\text{in }{\mathbb{T}}^{2}\Bigg{\\}}$ (0.2)

endowed with the $L^{2}$-scalar product $\langle\cdot,\cdot\rangle$ and the
corresponding norm $\|\cdot\|.$ Projecting the system (0.1) to the space $H$,
we eliminate the pressure term and obtain the evolution equation

$\partial_{t}u-\nu\Delta u+\Pi(\langle u,\nabla\rangle u)=\Pi\eta,$ (0.3)

where $\Pi$ is the Leray projection to $H$ in
$L^{2}({\mathbb{T}}^{2},{\mathbb{R}}^{2})$ (see Section 6 in Chapter 1 of
[Lio69]). We assume that $\eta$ is a white-in-time noise of the form

$\eta(t,x)=\partial_{t}W(t,x),\quad W(t,x)=\sum_{l\in{\cal
K}}b_{l}W_{l}(t)e_{l}(x),$ (0.4)

where ${\cal K}\subset{\mathbb{Z}}^{2}_{*}$ is a finite set,
$\\{b_{l}\\}_{l\in{\cal K}}$ are non-zero real numbers,
$\\{W_{l}\\}_{l\in{\cal K}}$ are independent standard Brownian motions on a
filtered probability space $(\Omega,{\cal F},\\{{\cal F}_{t}\\},{\mathbb{P}})$
satisfying the usual conditions (see Definition 2.25 in [KS91]), and

$e_{l}(x)=\begin{cases}l^{\bot}\cos\langle l,x\rangle&\text{if }l_{1}>0\text{
or }l_{1}=0,\,l_{2}>0,\\\ l^{\bot}\sin\langle l,x\rangle&\text{if
}l_{1}<0\text{ or }l_{1}=0,\,l_{2}<0,\quad l=(l_{1},l_{2})\end{cases}$

with $l^{\bot}=(-l_{2},l_{1}).$ In other words, ${\cal K}$ is the collection
of the Fourier modes directly perturbed by the noise. Under the above
assumptions, the NS system (0.3) defines a family of Markov processes
$(u_{t},{\mathbb{P}}_{u})$ parametrised by the initial condition $u(0)=u\in
H$. The ergodic properties of this family have been extensively studied in the
literature. It is now well known that $(u_{t},{\mathbb{P}}_{u})$ admits a
unique and exponentially mixing stationary measure, provided that the set
${\cal K}$ is sufficiently large. Under the condition that ${\cal K}$ contains
all the determining modes, the ergodicity has been established in different
settings in the papers [FM95, KS00, EMS01, KS02, BKL02]. Later, it was shown
that the ergodicity remains true for much smaller set ${\cal K}$; see the
papers [HM06, HM11, FGRT15] for the case when the noise is white-in-time and
[KNS20a, KNS20b] for the case of a general bounded noise. The reader is
referred to the book [KS12] for more references and for detailed description
of different methods.

In this paper, we study the Donsker–Varadhan type LDP for the NS system (0.3).
This type of LDP has been extensively studied in the case of finite-
dimensional diffusions and Markov processes in compact spaces; see the papers
[DV75], the books [FW84, DS89, DZ00], and the references therein. The paper
[Wu01] established a general criterion for Donsker–Varadhan type LDP for
Markov processes that are strong Feller and irreducible. In that paper the
criterion is applied to a class of stochastic damping Hamiltonian systems.
There are only few papers considering the problem of LDP for randomly forced
PDEs. The first results are obtained in [Gou07a, Gou07b] in the case of the
stochastic Burgers and NS equations with strong assumptions on the decay of
the coefficients $\\{b_{l}\\}$. Indeed, these papers use the criterion of
[Wu01], so they require some lower bounds for $\\{b_{l}\\}$ in order to
guarantee the strong Feller property. These assumptions have been relaxed to
the conditions $b_{l}\neq 0$ for all $l\in{\mathbb{Z}}^{2}_{*}$ and
$\sum_{l\in{\mathbb{Z}}^{2}_{*}}|l||b_{l}|^{2}<+\infty$ in the papers [JNPS15,
JNPS18], where a family of dissipative PDEs is considered driven by a random
kick-force. The proofs of these papers are based on a study of the long-time
behaviour of Feynman–Kac semigroup and a Kifer type criterion for the LDP.
Under similar non-degeneracy conditions, the local LDP is proved in [MN18] for
the stochastic damped nonlinear wave equation, and the full LDP is proved in
[Ner19] for the stochastic NS system. A controllability approach is used in
[JNPS21] to prove the LDP for the Lagrangian trajectories of the NS system.
Recently, the criterion of [Wu01] has been used in [WX18] in the case of SPDEs
driven by stable type noises and in [WXX21] in the case of non-linear monotone
SPDEs with white-in-time noise.

All the papers mentioned above establish the LDP under the assumption that the
noise is non-degenerate, i.e., perturbs directly all the Fourier modes in the
equation. The goal of the present paper is to establish the LDP in the case of
a highly degenerate noise, i.e., when only few Fourier modes are directly
perturbed. To formulate our main result, let us recall that a set ${\cal
K}\subset{\mathbb{Z}}^{2}_{*}$ is a generator if any element of
${\mathbb{Z}}^{2}$ is a finite linear combination of elements of ${\cal K}$
with integer coefficients. In what follows, we assume that the following
condition is satisfied.

(H) The set ${\cal K}\subset{\mathbb{Z}}^{2}_{*}$ in (0.4) is a finite
symmetric (i.e., $-{\cal K}={\cal K}$) generator that contains at least two
non-parallel vectors $m$ and $n$ such that $|m|\neq|n|$.

This is the condition under which the ergodicity of the NS system is
established in [HM06, HM11] in the case of a white-in-time noise and in
[KNS20a] in the case of a bounded noise. The set

${\cal K}=\\{(1,0),\,(-1,0),\,(1,1),\,(-1,-1)\\}\subset{\mathbb{Z}}^{2}_{*}$

is an example satisfying this condition.

For any $u\in H$, let us define the family of occupation measures

$\zeta_{t}=\frac{1}{t}\int_{0}^{t}\delta_{u_{s}}{\textup{d}}s,\,\,\,t>0$ (0.5)

on the probability space $(\Omega,{\cal F},{\mathbb{P}}_{u})$, where
$\delta_{v}$ is the Dirac measure concentrated at $v\in H$.

######  Main Theorem.

Under the Condition (H), the family $\\{\zeta_{t},t>0\\}$ satisfies the LDP.

See Theorem 1.1 for more detailed formulation of this result. The proof is
carried out by using a criterion for the LDP developed in [JNPS18] in a
discrete-time setting and extended in [MN18] to the continuous-time. According
to that criterion, the LDP will be established if we show that the following
five properties hold for the Feynman–Kac semigroup associated with the NS
system (0.3): growth properties, existence of eigenvector, time-continuity,
uniform irreducibility, and uniform Feller property. The first three
properties are verified in [Ner19] and they hold no matter how degenerate is
the noise. The uniform irreducibility property follows from the approximate
controllability results obtained in [AS05, AS06]. It is interesting to note
that Condition (H) is necessary and sufficient for the approximate
controllability of the NS system if one uses controls acting via the Fourier
modes in ${\cal K}$. The main technical difficulty of this paper is related to
the verification of the uniform Feller property, which we carry out by
developing the Malliavin calculus analysis of the papers [MP06, HM06, HM11].
More precisely, we derive the uniform Feller property from a gradient estimate
for the Feynman–Kac semigroup. The proof of latter contains essential
differences with respect to the situations studied in [MP06, HM06, HM11]
because of the non-Markovian character of the Feynman–Kac semigroup.

This paper is organised as follows. In Section 1, we explain how the Main
Theorem is derived from the above-mentioned five properties. In Section 2, we
recall some elements of Malliavin calculus, and in Section 3, we verify the
uniform Feller property.

#### Acknowledgement

Xuhui Peng would like to gratefully thank University of Macau for the
hospitality, his research is supported in part by by NNSFC (No. 12071123),
Scientific Research Project of Hunan Province Education Department (No.
20A329) and Program of Constructing Key Discipline in Hunan Province. Lihu Xu
is supported in part by NNSFC grant (No.12071499), Macao S.A.R grant FDCT
0090/2019/A2 and University of Macau grant MYRG2018-00133-FST.

#### Notation

In this paper, we use the following notation.

$H$ is the space of divergence-free square-integrable vector fields on
${\mathbb{T}}^{2}$ with zero mean value (see (0.2)). It is endowed with the
$L^{2}$-norm $\|\cdot\|$.

$H^{m}=H^{m}({\mathbb{T}}^{2},{\mathbb{R}}^{2})\cap H$, where
$H^{m}({\mathbb{T}}^{2},{\mathbb{R}}^{2})$ is the Sobolev space of order
$m\geq 1$. We endow the space $H^{m}$ with the usual Sobolev norm
$\|\cdot\|_{m}$.

$B_{H^{m}}(a,r)$ is the closed ball in $H^{m}$ of radius $r>0$ centred at $a$.
We write $B_{H^{m}}(r)$ when $a=0$.

We consider the NS system in the vorticity formulation in the space of square
integrable zero mean functions:

$\displaystyle\tilde{H}=\left\\{w\in
L^{2}({\mathbb{T}}^{2},{\mathbb{R}}):\int_{{\mathbb{T}}^{2}}w(x){\textup{d}}x=0\right\\}$
(0.6)

equipped with the $L^{2}$-norm $\|\cdot\|.$ Let
$\tilde{H}^{m}=H^{m}({\mathbb{T}}^{2},{\mathbb{R}})\cap\tilde{H}$, $m\geq 1$
be endowed with the Sobolev norm denoted by $\|\cdot\|_{m}$.

$L^{\infty}(H)$ is the space of bounded Borel-measurable functions
$\psi:H\to{\mathbb{R}}$ with the norm $\|\psi\|_{\infty}=\sup_{u\in
H}|\psi(u)|$. $C_{b}(H)$ is the space of continuous functions $\psi\in
L^{\infty}(H)$. $C^{1}_{b}(H)$ is the space of functions $\psi\in C_{b}(H)$
that are continuously Fréchet differentiable with bounded derivative.

Let ${\mathfrak{w}}:H\to[1,+\infty]$ be a Borel-measurable function. Then
$C_{\mathfrak{w}}(H)$ (resp., $L_{\mathfrak{w}}^{\infty}(H)$) is the space of
continuous (resp., Borel-measurable) functions $\psi:H\to{\mathbb{R}}$ such
that $\|\psi\|_{L_{\mathfrak{w}}^{\infty}}=\sup_{u\in
H}|\psi(u)|/{\mathfrak{w}}(u)<+\infty.$

${\cal M}_{+}(H)$ is the collection of non-negative finite Borel measures on
$H$ endowed with the weak convergence topology. For any $\psi\in
L^{\infty}(H)$ and $\mu\in{\cal M}_{+}(H)$, we write
$\langle\psi,\mu\rangle=\int_{H}\psi(u)\mu({\textup{d}}u).$ ${\cal P}(H)$ is
the subset of probability measures, and ${\cal P}_{\mathfrak{w}}(H)$ is the
set of $\mu\in{\cal P}(H)$ such that
$\langle{\mathfrak{w}},\mu\rangle<+\infty$.

${\mathcal{L}}(X,Y)$ is the space of linear bounded operators between Banach
spaces $X$ and $Y$ endowed with the natural norm
$\|\cdot\|_{{\mathcal{L}}(X,Y)}$.

The letter $C$ is used to denote unessential constants that can change from
line to line.

##  1 Main results

###  1.1 LDP and multiplicative ergodicity

Recall that a mapping $I:{\cal P}(H)\to[0,+\infty]$ is a good rate function if
its level sets

$\left\\{\sigma\in{\cal P}(H):I(\sigma)\leq\alpha\right\\},\quad\alpha\geq 0$

are compact. Moreover, if the effective domain of $I$, defined by

$D_{I}=\left\\{\sigma\in{\cal P}(H):I(\sigma)<+\infty\right\\},$

is not a singleton, we say that $I$ is a non-trivial good rate function. For
any $\gamma>0$ and $M>0$, we set

$\Lambda(\gamma,M):=\Bigg{\\{}\nu\in{\cal
P}(H):\int_{H}e^{\gamma\|u\|^{2}}\nu({\textup{d}}u)\leq M\Bigg{\\}}.$

The following is a more detailed version of the Main Theorem formulated in the
Introduction.

######  Theorem 1.1.

Assume that Condition (H) is verified. Then, for any $\gamma>0$ and $M>0$, the
family of random probability measures $\\{\zeta_{t},t>0\\}$ defined by (0.5)
satisfies the LDP uniformly w.r.t. the initial measure
$\nu\in\Lambda(\gamma,M)$. More precisely, there is a non-trivial good rate
function $I:{\cal P}(H)\to[0,+\infty]$ that does not depend on $\gamma$ and
$M$ and satisfies the inequalities

$\displaystyle\limsup_{t\to+\infty}\frac{1}{t}\log\sup_{\nu\in\Lambda(\gamma,M)}{\mathbb{P}}_{\nu}\left\\{\zeta_{t}\in
F\right\\}\leq-\inf_{\sigma\in F}I(\sigma),$
$\displaystyle\liminf_{t\to+\infty}\frac{1}{t}\log\inf_{\nu\in\Lambda(\gamma,M)}{\mathbb{P}}_{\nu}\left\\{\zeta_{t}\in
G\right\\}\geq-\inf_{\sigma\in G}I(\sigma)$

for any closed set $F$ and any open set $G$ in ${\cal P}(H)$.

This theorem is derived from a multiplicative ergodic theorem for the NS
system. To formulate that result, let us introduce the following weight
functions:

$\displaystyle{\mathfrak{m}}_{\gamma}(u)$
$\displaystyle=\exp\left(\gamma\|u\|^{2}\right),\quad\gamma>0,$
$\displaystyle{\mathfrak{w}}_{m}(u)$ $\displaystyle=1+\|u\|^{2m},\quad m\geq
1,\,\,u\in H.$

There is a constant $\gamma_{0}=\gamma_{0}({\mathfrak{B}}_{0})>0$, where
${\mathfrak{B}}_{0}=\sum_{l\in{\cal K}}b_{l}^{2}$, such that

$\displaystyle{\mathbb{E}}_{u}{\mathfrak{m}}_{\gamma}(u_{t})$
$\displaystyle\leq e^{-\gamma t}{\mathfrak{m}}_{\gamma}(u)+C,$ (1.1)
$\displaystyle{\mathbb{E}}_{u}{\mathfrak{w}}_{m}(u_{t})$ $\displaystyle\leq
e^{-2mt}{\mathfrak{w}}_{m}(u)+C$ (1.2)

for any $\gamma\in(0,\gamma_{0})$, $m\geq 1$, $u\in H$, and $t\geq 0$, where
$C=C(m,\varkappa,{\mathfrak{B}}_{0})>0$ is a constant; e.g., see Proposition
2.4.9 in [KS12] and Lemma 5.3 in [Ner19] for a proof of these inequalities.

For any $V\in C_{b}(H),$ the Feynman–Kac semigroup associated with the Markov
family $(u_{t},{\mathbb{P}}_{u})$ is defined by

${\mathfrak{P}}_{t}^{V}\psi(u)={\mathbb{E}}_{u}\left\\{\exp\left(\int_{0}^{t}V(u_{s}){\textup{d}}s\right)\psi(u_{t})\right\\},\quad{\mathfrak{P}}_{t}^{V}:C_{b}(H)\to
C_{b}(H);$

its dual is denoted by ${\mathfrak{P}}_{t}^{V*}:{\cal M}_{+}(H)\to{\cal
M}_{+}(H)$. From (1.1) it follows that ${\mathfrak{P}}_{t}^{V}$ maps the space
$C_{{\mathfrak{m}}_{\gamma}}(H)$ into itself for $\gamma\in(0,\gamma_{0})$.

######  Theorem 1.2.

Assume that Condition (H) is verified and $V\in C_{b}^{1}(H)$. Then there are
constants $m=m(V)\geq 1$ and
$\gamma=\gamma({\mathfrak{B}}_{0})\in(0,\gamma_{0})$ such that there are
unique eigenvectors $h_{V}\in C_{{\mathfrak{w}}_{m}}(H)$ and $\mu_{V}\in{\cal
P}_{{\mathfrak{m}}_{\gamma}}(H)$ for the semigroups ${\mathfrak{P}}_{t}^{V}$
and ${\mathfrak{P}}_{t}^{V*}$ corresponding to an eigenvalue $\lambda_{V}>0$,
i.e,

${\mathfrak{P}}^{V*}_{t}\mu_{V}=\lambda_{V}^{t}\mu_{V},\quad\quad{\mathfrak{P}}^{V}_{t}h_{V}=\lambda_{V}^{t}h_{V}\quad\quad\text{for
$t>0$},$

and normalised by $\langle h_{V},\mu_{V}\rangle=1$. For any $\psi\in
C_{{\mathfrak{m}}_{\gamma}}(H),\,\nu\in{\cal P}(H)$, and $R>0$, the following
limits hold as $t\to+\infty$:

$\displaystyle\lambda_{V}^{-t}{\mathfrak{P}}_{t}^{V}\psi\to\,$
$\displaystyle\langle\psi,\mu_{V}\rangle h_{V}\,\,\textup{in
}C_{b}(B_{H}(R))\cap L^{1}(H,\mu_{V}),$
$\displaystyle\lambda^{-t}_{V}{\mathfrak{P}}_{t}^{V*}\nu\to\,$
$\displaystyle\langle h_{V},\nu\rangle\mu_{V}\,\,\textup{in }{\cal M}_{+}(H).$

Furthermore, for any $M>0$ and $\varkappa\in(0,\gamma)$,

$\lambda_{V}^{-t}\,{\mathbb{E}}_{\nu}\left\\{\exp\bigg{(}\int_{0}^{t}V(u_{s}){\textup{d}}s\bigg{)}\psi(u_{t})\right\\}\to\langle\psi,\mu_{V}\rangle\,\langle
h_{V},\nu\rangle$

uniformly w.r.t. $\nu\in\Lambda(\varkappa,M)$ as $t\to+\infty$.

This theorem improves Theorem 1.1 in [Ner19] in two directions. First, in this
theorem, the noise is very degenerate, while in [Ner19] all the Fourier modes
are assumed to be directly perturbed by the noise. Second, in the present
situation, the class of functions $V$ is larger, since the result in [Ner19]
applies only to functions depending on finite-dimensional projection of $u$.

Theorem 1.2 can be viewed as an improvement of Theorem 2.1 in [HM06]. Indeed,
in the case $V=0$, the Feynman–Kac semigroup reduces to the Markov semigroup
with eigenvalue $\lambda_{V}=1$, eigenvector $h_{V}={\mathbf{1}}$ (the
function identically equal to $1$ on $H$), and the measure $\mu_{V}=\mu$ is
the unique stationary measure. The above limits imply that $\mu$ is mixing.

Theorem 1.1 is derived from Theorem 1.2 by using a Kifer type criterion in
unbounded spaces. Since this derivation is literally the same as in the non-
degenerate case (see Section 1 in [Ner19]), we do not give the details. The
proof of Theorem 1.2 is discussed in the next subsection.

###  1.2 Proof of Theorem 1.2

The proof of Theorem 1.2 is carried out by applying a result on large-time
asymptotics of generalised Markov semigroups established in [JNPS18] in the
discrete-time setting and extended in [MN18] to the continuous-time. Here we
apply that result to the Feynman–Kac semigroup ${\mathfrak{P}}_{t}^{V}$ and
the associated kernel
$P_{t}^{V}(u,\Gamma)=({\mathfrak{P}}_{t}^{V*}\delta_{u})(\Gamma)$, $u\in H$,
$\Gamma\in{\mathcal{B}}(H),$ where $\delta_{u}$ is the Dirac measure
concentrated at $u$.

By the regularising property of the NS system, the measure
$P^{V}_{t}(u,\cdot)$ is concentrated on the space $H^{2}$ for any $u\in H$ and
$t>0$. For any $R>0$, let us denote $X_{R}=B_{H^{2}}(R)$, and let $V\in
C_{b}(H)$ be arbitrary. Then the following properties hold.

Growth properties.

There are numbers $R_{0}>0$, $\gamma\in(0,\gamma_{0})$, and $m\geq 1$ such
that the following quantities are finite:

$\sup_{t\geq
0}\frac{\|{\mathfrak{P}}^{V}_{t}{\mathfrak{w}}_{m}\|_{L_{{\mathfrak{w}}_{m}}^{\infty}}}{\|{\mathfrak{P}}^{V}_{t}{{\mathbf{1}}}\|_{R_{0}}},\,\,\,\,\sup_{t\geq
0}\frac{\|{\mathfrak{P}}_{t}^{V}{\mathfrak{m}}_{\gamma}\|_{L^{\infty}_{{\mathfrak{m}}_{\gamma}}}}{\|{\mathfrak{P}}_{t}^{V}{{\mathbf{1}}}\|_{R_{0}}},\,\,\,\,\sup_{t\geq
1}\frac{\|{\mathfrak{P}}_{t}^{V}\Phi\|_{L^{\infty}_{{\mathfrak{m}}_{\gamma}}}}{\|{\mathfrak{P}}_{t}^{V}{{\mathbf{1}}}\|_{R_{0}}},$
(1.3)

where $\|\psi\|_{R}=\sup_{u\in X_{R}}|\psi(u)|$ and
$\Phi(u)=\|u\|_{H^{2}}^{2}$.

Existence of an eigenvector.

For any $t>0$, there is a measure $\mu_{t,V}\in{\cal P}(H)$ and a number
$\lambda_{t,V}>0$ such that
${\mathfrak{P}}_{t}^{V*}\mu_{t,V}=\lambda_{t,V}\mu_{t,V}$. Moreover, for any
numbers $\varkappa\in(0,\gamma_{0})$ and $n,m\geq 1$, we have

$\displaystyle\int_{H}\\!\left(\|u\|_{H^{2}}^{n}+{\mathfrak{m}}_{\varkappa}(u)\right)\mu_{t,V}({\textup{d}}u)<+\infty,$
$\displaystyle\|{\mathfrak{P}}_{t}^{V}{\mathfrak{w}}_{m}\|_{X_{R}}\int_{X_{R}^{c}}\\!{\mathfrak{w}}_{m}(u)\mu_{t,V}({\textup{d}}u)\to
0\,\,\textup{as }R\to+\infty.$

Time-continuity.

For any $m\geq 1$, $\psi\in C_{{\mathfrak{w}}_{m}}(H)$, and $u\in H$, the
function $t\mapsto{\mathfrak{P}}^{V}_{t}\psi(u)$,
${\mathbb{R}}_{+}\to{\mathbb{R}}$ is continuous.

Uniform irreducibility.

For any $\rho,r,R>0$, there are numbers $l=l(\rho,r,R)>0$ and
$p=p(V,\rho,r)>0$ such that

$P^{V}_{l}(u_{0},B_{H}(\hat{u},r))\geq p$ (1.4)

for any $u_{0}\in X_{R}$ and $\hat{u}\in X_{\rho}$.

Uniform Feller property.

The family of functions
$\\{\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R}^{-1}{\mathfrak{P}}_{t}^{V}\psi,t\geq
0\\}$ is uniformly equicontinuous 555By uniform equicontinuity of
$\\{\|{\mathfrak{P}}_{t}^{V}{{\mathbf{1}}}\|_{R}^{-1}{\mathfrak{P}}^{V}_{t}\psi,t\geq
0\\}$ on $X_{R}$ we mean that for any $\varepsilon>0$, there is $\delta>0$
such that
$\|{\mathfrak{P}}^{V}_{t}{{\mathbf{1}}}\|_{R}^{-1}\big{|}{\mathfrak{P}}_{t}^{V}\psi(u)-{\mathfrak{P}}_{t}^{V}\psi(u^{\prime})\big{|}<\varepsilon$
for any $u,u^{\prime}\in X_{R}$ with $\|u-u^{\prime}\|<\delta$ and any $t\geq
0$.  on $X_{R}$ for any $V,\psi\in C_{b}^{1}(H)$ and $R\geq R_{0}.$

The first three of the above properties are established 666These propositions
and lemma in [Ner19] are formulated in the case when $X_{R}=B_{H^{1}}(R)$ and
the noise is non-degenerate. However, their proofs work in the setting of the
present paper without any change. in Propositions 2.1 and 2.5 and Lemma 2.3 in
[Ner19]. The proof of the uniform irreducibility is given below, and the
uniform Feller property is established in Section 3. Theorem 1.2 is obtained
by applying Theorem 7.4 in [MN18] and by literally repeating the arguments of
Section 4 in [Ner19].

###### Proof of uniform irreducibility.

Let $P_{t}(u_{0},\cdot)$ be the Markov transition kernel of the family
$(u_{t},{\mathbb{P}}_{u_{0}})$. The boundedness of $V$ implies that

$P_{t}^{V}(u_{0},{\textup{d}}v)\geq
e^{-t\|V\|_{\infty}}P_{t}(u_{0},{\textup{d}}v)\quad\textup{for }t>0,\,u_{0}\in
H.$ (1.5)

According to [AS05, AS06], under Condition (H), the NS system is approximately
controllable in the space $H$ by controls taking values in the space

${\cal H}_{\cal K}=\mathop{\rm span}\nolimits\\{e_{l}:l\in{\cal K}\\}.$

This implies that, for any $u_{0},\hat{u}\in H$ and $r>0$, there is a function
$\zeta\in C^{\infty}([0,1];{\cal H}_{\cal K})$ such that

$\|u(1,u_{0},\zeta)-\hat{u}\|<r,$

where $u(t,u_{0},\zeta)$ is the solution of the deterministic NS system (0.3)
with the initial condition $u(0)=u_{0}$ and the (control) force
$\eta=\partial_{t}\zeta$. Using the fact that the mapping
$(u_{0},\zeta)\mapsto u(1,u_{0},\zeta)$ is continuous from $H\times
C([0,1];{\cal H}_{\cal K})$ to $H$, the non-degeneracy of the law of the
Wiener process $W$ in $C([0,1];{\cal H}_{\cal K})$ (i.e., the support of the
law of $W$ coincides with the entire space $C([0,1];{\cal H}_{\cal K})$), a
simple compactness argument, and inequality (1.5), we arrive at (1.4). ∎

##  2 Elements of Malliavin calculus

The uniform Feller property is proved by using Malliavin calculus analysis
from the papers [MP06, HM06, HM11]. In this section, we recall some basic
definitions and estimates from there. To match the framework of these papers,
we rewrite the NS system (0.3) in the vorticity formulation:

$\partial_{t}w-\nu\Delta w+B({\cal K}w,w)=\sum_{l\in{\cal
K}}b_{l}|l|^{2}\dot{W}_{l}(t)\phi_{l},$ (2.1)

where $w=\nabla\wedge u$, $B(u,w)=\langle u,\nabla\rangle w$, and ${\cal K}$
is the Biot–Savart operator

${\cal K}w=\sum_{l\in{\mathbb{Z}}_{*}^{2}}|l|^{-2}l^{\bot}w_{-l}\phi_{l}$

with $|l|^{2}=l_{1}^{2}+l_{2}^{2}$, $l^{\bot}=(-l_{2},l_{1})$, $w_{l}=\langle
w,\phi_{l}\rangle$, and

$\phi_{l}(x)=\begin{cases}\sin\langle l,x\rangle&\text{if }l_{1}>0\text{ or
}l_{1}=0,\,l_{2}>0,\\\ -\cos\langle l,x\rangle&\text{if }l_{1}<0\text{ or
}l_{1}=0,\,l_{2}<0,\quad l=(l_{1},l_{2}).\end{cases}$

The operator ${\cal K}$ is continuous from
$H^{s}({\mathbb{T}}^{2};{\mathbb{R}})$ to
$H^{s+1}({\mathbb{T}}^{2};{\mathbb{R}}^{2})$ for any $s\in{\mathbb{R}}$; it
allows to recover the velocity field from the vorticity via $u={\cal K}w$.

We consider Eq. (2.1) in the space $\tilde{H}$ of real-valued square-
integrable functions on $\mathbb{T}^{2}$ with zero mean value (see (0.6)); it
is endowed with the $L^{2}$ norm $\|\cdot\|.$ Since the underlying probability
space plays no role, without loss of generality, we can assume that $\Omega$
is the Wiener space, $W(t)=\\{W_{l}(t)\\}_{l\in{\cal K}}$ is the canonical
process, and ${\mathbb{P}}$ is the Wiener measure. Furthermore, we denote by
$\\{\theta_{l}\\}_{l\in{\cal K}}$ the standard basis in ${\mathbb{R}}^{d}$
with $d=|{\cal K}|$, and define a linear map $Q:{\mathbb{R}}^{d}\to\tilde{H}$
by $Q\theta_{l}=b_{l}|l|^{2}\phi_{l}.$ Let $w_{t}=\Phi(t,w,W_{\cdot})$ be the
solution of Eq. (2.1) with initial value $w(0)=w\in\tilde{H}$. For any $0\leq
s\leq t$ and $\xi\in\tilde{H}$, let $J_{s,t}\xi$ be the solution of the
linearised problem:

$\displaystyle\partial_{t}J_{s,t}\xi-\nu\Delta
J_{s,t}\xi+\tilde{B}(w_{t},J_{s,t}\xi)$ $\displaystyle=0,$ (2.2)
$\displaystyle J_{s,s}\xi$ $\displaystyle=\xi,$

where $\tilde{B}(w,v)=B({\cal K}w,v)+B({\cal K}v,w)$.

Recall that, for given $T>0$ and $v\in L^{2}([0,T];{\mathbb{R}}^{d}),$ the
Malliavin derivative of $w_{t}$ in the direction $v$ is defined by

${\mathcal{D}}^{v}w_{t}=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\left(\Phi(t,w_{0},W+\varepsilon\int_{0}^{\cdot}v{\textup{d}}s)-\Phi(t,w_{0},W)\right),\quad$

where the limit holds almost surely (e.g., see the book [Nua06] for finite-
dimensional setting or the papers [MP06, HM06, HM11, FGRT15] for Hilbert space
case). By the Riesz representation theorem, there is a linear operator
${\mathcal{D}}:L^{2}(\Omega,\tilde{H})\to
L^{2}(\Omega;L^{2}([0,T];{\mathbb{R}}^{d})\otimes\tilde{H})$ such that

${\mathcal{D}}^{v}w=\langle{\mathcal{D}}w,v\rangle_{L^{2}([0,T];{\mathbb{R}}^{d})}.$
(2.3)

On the other hand, we have

${\mathcal{D}}^{v}w_{t}={\mathcal{A}}_{0,t}v,$ (2.4)

where ${\mathcal{A}}_{s,t}:L^{2}([s,t];{\mathbb{R}}^{d})\to\tilde{H}$ is the
random operator defined by

${\mathcal{A}}_{s,t}v=\int_{s}^{t}J_{r,t}Qv(r){\textup{d}}r,\quad 0\leq s\leq
t\leq T,$ (2.5)

i.e., ${\mathcal{A}}_{s,t}v$ is the solution of the linearised problem with a
source term:

$\displaystyle\partial_{t}{\mathcal{A}}_{s,t}v-\nu\Delta{\mathcal{A}}_{s,t}v+\tilde{B}(w_{t},{\mathcal{A}}_{s,t}v)$
$\displaystyle=Qv,$ $\displaystyle{\mathcal{A}}_{s,s}v$ $\displaystyle=0.$

The adjoint ${\mathcal{A}}_{s,t}^{*}:\tilde{H}\to
L^{2}([s,t];{\mathbb{R}}^{d})$ is given by

$({\mathcal{A}}_{s,t}^{*}\xi)(r)=Q^{*}J_{r,t}^{*}\xi,\quad\xi\in\tilde{H},\,r\in[s,t],$

where $Q^{*}:\tilde{H}\rightarrow{\mathbb{R}}^{d}$ is the adjoint of $Q$.

Let us denote by $J^{(2)}_{s,t}(\phi,\psi)$ the second derivative of $w_{t}$
with respect to $w$ in the directions of $\phi$ and $\psi$. It is the solution
of the problem

$\displaystyle\partial_{t}J^{(2)}_{s,t}(\phi,\psi)-\nu\Delta
J^{(2)}_{s,t}(\phi,\psi)+\tilde{B}(J_{s,t}\phi,J_{s,t}\psi)+\tilde{B}(w_{t},J^{(2)}_{s,t}(\phi,\psi))$
$\displaystyle=0,$ $\displaystyle J^{(2)}_{s,s}(\phi,\psi)$ $\displaystyle=0.$

The next lemma follows from Lemma 4.10 in [HM06].

######  Lemma 2.1.

For any $\kappa,p>0$, $0\leq\tau<T$, and $w\in\tilde{H}$, we have

$\displaystyle{\mathbb{E}}_{w}\sup_{s<t\in[\tau,T]}\|J_{s,t}\|^{p}_{{\mathcal{L}}(\tilde{H},\tilde{H})}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}\\},$ (2.6)
$\displaystyle{\mathbb{E}}_{w}\sup_{s<t\in[\tau,T]}|||J^{(2)}_{s,t}|||^{p}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}\\},$

where $|||J^{(2)}_{s,t}|||=\sup_{\|\phi\|,\|\psi\|\leq
1}\|J^{(2)}_{s,t}(\phi,\psi)\|$, and $C=C(\kappa,p,T-\tau,\mathfrak{B}_{0})>0$
is a constant.

For any $0\leq s<t,$ the Malliavin operator is defined by

${\mathcal{M}}_{s,t}={\mathcal{A}}_{s,t}{\mathcal{A}}_{s,t}^{*}:\tilde{H}\rightarrow\tilde{H}.$

It is a non-negative self-adjoint operator, so its regularisation
${\mathcal{M}}_{s,t}+\beta{\mathbb{I}}$ is invertible for any $\beta>0$. Here
${\mathbb{I}}$ is the identity. The following lemma gathers some estimates
from Section 4.8 in [HM06] and Lemma A.6 in [FGRT15].

######  Lemma 2.2.

There is a constant $C=C(\mathfrak{B}_{0})>0$ such that, for any $0\leq s<t$,
$\beta>0$, and $w\in\tilde{H}$, we have

$\displaystyle\|{\mathcal{A}}_{s,t}\|^{2}_{{\mathcal{L}}(L^{2}([s,t];{\mathbb{R}}^{d}),\tilde{H})}\leq
C\int_{s}^{t}\|J_{r,t}\|_{{\mathcal{L}}(\tilde{H},\tilde{H})}^{2}{\textup{d}}r,$
(2.7)
$\displaystyle\|{\mathcal{A}}_{s,t}^{*}({\mathcal{M}}_{s,t}+\beta{\mathbb{I}})^{-1/2}\|_{{\mathcal{L}}(\tilde{H},L^{2}([s,t];{\mathbb{R}}^{d}))}\leq
1,$ (2.8)
$\displaystyle\|({\mathcal{M}}_{s,t}+\beta{\mathbb{I}})^{-1/2}{\mathcal{A}}_{s,t}\|_{{\mathcal{L}}(L^{2}([s,t];{\mathbb{R}}^{d}),\tilde{H})}\leq
1,$ (2.9)
$\displaystyle\|({\mathcal{M}}_{s,t}+\beta{\mathbb{I}})^{-1/2}\|_{{\mathcal{L}}(\tilde{H},\tilde{H})}\leq\beta^{-1/2}.$
(2.10)

We shall use the notation

${\mathcal{D}}_{r}F=({\mathcal{D}}F)(r),\quad{\mathcal{D}}^{j}F=({\mathcal{D}}F)^{j},\quad{\mathcal{D}}_{r}^{j}F=({\mathcal{D}}F)^{j}(r),\quad
j=1,\ldots,d.$

From the equalities (2.3)-(2.5) it follows that
${\mathcal{D}}_{r}^{i}w_{t}=J_{r,t}Q\theta_{i}$, $0\leq r\leq t$. From this
and (2.2), we conclude that, for $0\leq s<t$,

$\partial_{t}{\mathcal{D}}_{r}^{i}J_{s,t}\xi-\nu\Delta{\mathcal{D}}_{r}^{i}J_{s,t}\xi+\tilde{B}(w_{t},{\mathcal{D}}_{r}^{i}J_{s,t}\xi)+\tilde{B}(J_{r,t}Q\theta_{i},J_{s,t}\xi)=0.$

Furthermore, by the variation of constants formula, we have

${\mathcal{D}}_{r}^{i}J_{s,t}\xi=\begin{cases}J^{(2)}_{r,t}(Q\theta_{i},J_{s,r}\xi)&\text{
for }r\geq s,\\\ J^{(2)}_{s,t}(J_{r,s}Q\theta_{i},\xi)&\text{ for }r\leq
s.\end{cases}$

This equality and Lemma 2.1 imply the following lemma. For further details, we
refer the reader to Section 4.8 in [HM06] and Lemma A.7 in [FGRT15].

######  Lemma 2.3.

The operators $J_{s,t}$, ${\mathcal{A}}_{s,t}$, and ${\mathcal{A}}_{s,t}^{*}$
are Malliavin differentiable, and for any $\kappa>0$, $r\in[s,t]$, $p>0$, and
$w\in\tilde{H}$, the following inequalities hold

$\displaystyle{\mathbb{E}}_{w}\|{\mathcal{D}}_{r}^{i}J_{s,t}\|^{p}_{{\mathcal{L}}(\tilde{H},\tilde{H})}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}\\},$ (2.11)
$\displaystyle{\mathbb{E}}_{w}\|{\mathcal{D}}_{r}^{i}{\mathcal{A}}_{s,t}\|^{p}_{{\mathcal{L}}(L^{2}([s,t];{\mathbb{R}}^{d}),\tilde{H})}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}\\},$ (2.12)
$\displaystyle{\mathbb{E}}_{w}\|{\mathcal{D}}_{r}^{i}{\mathcal{A}}^{*}_{s,t}\|^{p}_{{\mathcal{L}}(\tilde{H},L^{2}([s,t];{\mathbb{R}}^{d}))}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}\\},$ (2.13)

where $C=C(\kappa,p,t-s,\mathfrak{B}_{0})>0$.

##  3 Proof of uniform Feller property

###  3.1 Reduction to a gradient estimate

The aim of this section is to prove the following proposition.

######  Proposition 3.1.

Under Condition (H), for any $V,\psi\in C_{b}^{1}(H)$, there is a number
$R_{0}=R_{0}(V)>0$ such that the family
$\left\\{\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R}^{-1}{\mathfrak{P}}_{t}^{V}\psi,t\geq
0\right\\}$ is uniformly equicontinuous on $X_{R}$ for any $R\geq R_{0}.$

###### Proof.

For any $V,\psi\in C_{b}^{1}(H),$ let us define functions
$\tilde{V},\tilde{\psi}\in C_{b}^{1}(\tilde{H})$ by $\tilde{V}(w)=V({\cal
K}w)$ and $\tilde{\psi}(w)=\psi({\cal K}w)$, $w\in\tilde{H}$. The Feynman–Kac
semigroup associated with Eq. (2.1) is given by

$\tilde{\mathfrak{P}}_{t}^{\tilde{V}}\tilde{\psi}(w)={\mathbb{E}}_{w}\left\\{\exp\left(\int_{0}^{t}\tilde{V}(w_{s}){\textup{d}}s\right)\tilde{\psi}(w_{t})\right\\},\quad\tilde{\mathfrak{P}}_{t}^{\tilde{V}}:C_{b}^{1}(\tilde{H})\to
C_{b}^{1}(\tilde{H}).$

In what follows, the number $R_{0}$ is chosen such that the growth properties
(1.3) hold. In the next subsection, we prove the following proposition (cf.
Proposition 4.3 in [HM06]).

######  Proposition 3.2.

Under the conditions of Proposition 3.1, for any numbers $\kappa>0$ and
$a\in(0,1)$, there is a constant $C=C(\kappa,a,\|\nabla
V\|_{\infty},\|V\|_{\infty})>0$ such that

$\|\nabla_{\xi}\tilde{{\mathfrak{P}}}_{t}^{\tilde{V}}\tilde{\psi}(w)\|\leq
C\exp\\{\kappa\|w\|^{2}\\}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}\left[\|\nabla\psi\|_{\infty}a^{t}+\|\psi\|_{\infty}\right]\|\xi\|$
(3.1)

for any $w,\xi\in\tilde{H}$ and $t\geq 0$. Here $\nabla_{\xi}$ is the
derivative with respect to the initial condition in the direction $\xi$.

This result implies Proposition 3.1. Indeed, let us take any $u_{1},u_{2}\in
X_{R}$ and set $w_{i}=\nabla\wedge u_{i}$, $i=1,2$. Using inequality (3.1)
with any $\kappa>0$ and $a\in(0,1)$ and an interpolation inequality, we see
that

$\displaystyle\big{|}{\mathfrak{P}}_{t}^{V}\psi(u_{1})-{\mathfrak{P}}_{t}^{V}\psi(u_{2})\big{|}$
$\displaystyle=\big{|}\tilde{{\mathfrak{P}}}_{t}^{\tilde{V}}\tilde{\psi}(w_{1})-\tilde{{\mathfrak{P}}}_{t}^{\tilde{V}}\tilde{\psi}(w_{2})\big{|}$
$\displaystyle\leq
C\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\|w_{1}-w_{2}\|$
$\displaystyle\leq
C\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\|u_{1}-u_{2}\|_{1}$
$\displaystyle\leq
C\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\|u_{1}-u_{2}\|^{1/2}\|u_{1}-u_{2}\|_{2}^{1/2}$
$\displaystyle\leq
C\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\|u_{1}-u_{2}\|^{1/2},$

where $C=C(R,\|V\|_{\infty},\|\nabla
V\|_{\infty},\|\psi\|_{\infty},\|\nabla\psi\|_{\infty})>0.$ This completes the
proof of Proposition 3.1. ∎

###  3.2 Proof of Proposition 3.2

Let us take any $\xi\in\tilde{H}$ with $\|\xi\|=1$, denote

$\Xi_{t}=\exp\left(\int_{0}^{t}\tilde{V}(w_{s}){\textup{d}}s\right),$

and compute the derivative of
$\tilde{\mathfrak{P}}_{t}^{\tilde{V}}\tilde{\psi}(w)$ with respect to $w$ in
the direction $\xi$:

$\nabla_{\xi}\tilde{\mathfrak{P}}_{t}^{\tilde{V}}\tilde{\psi}(w)={\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{0}^{t}\nabla\tilde{V}(w_{s})J_{0,s}\xi{\textup{d}}s+\Xi_{t}\nabla\tilde{\psi}(w_{t})J_{0,t}\xi\right].$
(3.2)

Inspired by the papers [HM06, HM11], the idea of the proof of Proposition 3.2
is to approximate the perturbation $J_{0,t}\xi$ caused by the perturbation
$\xi$ of the initial condition with a variation ${\mathcal{A}}_{0,t}v$ coming
from a variation of the noise by an appropriate process $v$. Let us denote by
$\rho_{t}$ the residual error between $J_{0,t}\xi$ and ${\mathcal{A}}_{0,t}v$:

$\rho_{t}=J_{0,t}\xi-{\mathcal{A}}_{0,t}v,$

replace the term $J_{0,t}\xi$ in (3.2) by ${\mathcal{A}}_{0,t}v+\rho_{t}$, and
recall that ${\mathcal{A}}_{0,t}v={\mathcal{D}}^{v}w_{t}$ is the Malliavin
derivative of $w_{t}$ in the direction $v$. Then, at least formally, using the
Malliavin chain rule (see Proposition 1.2.3 in [Nua06]), we have

$\displaystyle\nabla_{\xi}\tilde{\mathfrak{P}}_{t}^{\tilde{V}}\tilde{\psi}(w)$
$\displaystyle={\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{0}^{t}\nabla\tilde{V}(w_{s}){\mathcal{D}}^{v}w_{s}{\textup{d}}s+\Xi_{t}\nabla\tilde{\psi}(w_{t}){\mathcal{D}}^{v}w_{t}\right]$
$\displaystyle\quad+{\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{0}^{t}\nabla\tilde{V}(w_{s})\rho_{s}{\textup{d}}s+\Xi_{t}\nabla\tilde{\psi}(w_{t})\rho_{t}\right]$
$\displaystyle={\mathbb{E}}_{w}\left[{\mathcal{D}}^{v}\left(\Xi_{t}\tilde{\psi}(w_{t})\right)\right]+{\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{0}^{t}\nabla\tilde{V}(w_{s})\rho_{s}{\textup{d}}s\right]$
$\displaystyle\quad+{\mathbb{E}}_{w}\left[\Xi_{t}\nabla\tilde{\psi}(w_{t})\rho_{t}\right]=I_{1}+I_{2}+I_{3}.$
(3.3)

The term $I_{1}$ is treated using Malliavin integration by parts formula (see
Lemma 1.2.1 in [Nua06]):

$I_{1}={\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{0}^{t}v(s){\textup{d}}W(s)\right],$
(3.4)

where the stochastic integral $\int_{0}^{t}v(s){\textup{d}}W(s)$ is in the
Skorokhod sense. The goal is to choose the process $v$ in a such way that the
terms $I_{i}$, $i=1,2,3$ are bounded by the right-hand side of inequality
(3.1). We use the same choice of $v$ as in the papers [HM06, HM11]. More
precisely, for any integer $n\geq 0$, the restriction $v_{n,n+1}$ of the
process $v$ to the time interval $[n,n+1]$ is defined by

$v_{n,n+1}(t)=\begin{cases}\big{(}{\mathcal{A}}_{n,n+1/2}^{*}\left({\mathcal{M}}_{n,n+1/2}+\beta{\mathbb{I}}\right)^{-1}J_{n,n+1/2}\rho_{n}\big{)}(t),\,\,\,\,\quad
t\in[n,n+1/2],\\\
0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
t\in[n+1/2,n+1],\end{cases}$ (3.5)

where we set $\rho_{0}=\xi$ and $\beta>0$ is a small parameter. This choice
allows to have an exponential decay for the moments of $\rho_{t}$ and of the
Skorokhod integral as proved in the following lemmas. Inequality (3.1) is
proved by combining these lemmas (with an appropriate choice of parameters
therein) and using a growth property of the Feynman–Kac semigroup.

The following two lemmas are versions of Propositions 4.13 and 4.14 in [HM06].
Since their formulations differ from the original ones, we give rather
detailed proofs based on the estimates recalled in Section 2.

######  Lemma 3.3.

For any $\kappa>0$ and $\alpha>0$, there are constants
$\beta=\beta(\kappa,\alpha)>0$ and $C=C(\kappa,\alpha)>0$ such that

${\mathbb{E}}\|\rho_{t}\|^{4}\leq C\exp\left\\{\kappa\|w\|^{2}-\alpha
t\right\\}\quad\text{for any $w\in\tilde{H}$ and $t\geq 0$.}$ (3.6)

###### Proof.

For integer times, this result is established in Proposition 4.13 in [HM06]
(this is where Condition (H) is used). Therefore, there are
$\beta=\beta(\kappa,\alpha)>0$ and $C=C(\kappa,\alpha)>0$ such that

${\mathbb{E}}\|\rho_{n}\|^{4}\leq C\exp\\{\kappa\|w\|^{2}-\alpha
n\\}\quad\text{for any $w\in\tilde{H}$ and $n\geq 0$}.$ (3.7)

From the construction it follows that

$\rho_{t}=\begin{cases}J_{n,t}\rho_{n}-{\mathcal{A}}_{n,t}v_{n,t},&\text{for
}t\in[n,n+1/2],\\\ J_{n+1/2,t}\rho_{n+1/2},&\text{for
}t\in[n+1/2,n+1]\end{cases}$

for any $n\geq 0$. Using (3.5) and inequalities (2.8) and (2.10), we get

$\|v_{n,n+1/2}\|_{L^{2}([n,n+1/2];{\mathbb{R}}^{d})}\leq\beta^{-1/2}\|J_{n,n+1/2}\rho_{n}\|.$
(3.8)

Hence, for any $t\in[n,n+1/2]$,

$\displaystyle\|\rho_{t}\|$ $\displaystyle\leq\|J_{n,t}\rho_{n}\|+\|{\cal
A}_{n,t}v_{n,t}\|$ $\displaystyle\leq\|J_{n,t}\rho_{n}\|+\|{\cal
A}_{n,t}\|_{{\mathcal{L}}(L^{2}([n,t];{\mathbb{R}}^{d}),\tilde{H})}\|v_{n,t}\|_{L^{2}([n,n+1/2];{\mathbb{R}}^{d})}$
$\displaystyle\leq\|J_{n,t}\rho_{n}\|+\|{\cal
A}_{n,t}\|_{{\mathcal{L}}(L^{2}([n,t];{\mathbb{R}}^{d}),\tilde{H})}\|v_{n,n+1/2}\|_{L^{2}([n,n+1/2];{\mathbb{R}}^{d})}$
$\displaystyle\leq
C\Big{(}\|J_{n,t}\rho_{n}\|+\beta^{-1/2}\|J_{n,n+1/2}\rho_{n}\|\sup_{s\in[n,t]}\|J_{s,t}\|_{{\mathcal{L}}(\tilde{H},\tilde{H})}\Big{)},$
(3.9)

where we used (2.7) and (3.8). For any $t\in[n+1/2,n+1],$ it holds that

$\|\rho_{t}\|\leq\sup_{s\in[n+1/2,t]}\|J_{s,t}\rho_{n+1/2}\|.$

Combining this with inequalities (2.6), (3.7), (3.9), the Cauchy–Schwarz
inequality, and the fact that $\kappa>0$ and $\alpha>0$ are arbitrary, we
arrive at (3.6). ∎

######  Lemma 3.4.

The constants $\beta>0$ and $C>0$ in Lemma 3.3 can be chosen such that also

${\mathbb{E}}\left|\int_{n}^{t}v(s){\textup{d}}W(s)\right|^{2}\leq
C\exp\\{\kappa\|w\|^{2}-\alpha n\\}$ (3.10)

for any $n\geq 0$, $t\in[n,n+1]$, and $w\in\tilde{H}$.

###### Proof.

In this proof, we consider the endpoint case $t=n+1$; the case $t\in[n,n+1)$
is treated in a similar way. Using the generalised It$\hat{\text{o}}$ isometry
(see Section 1.3 in [Nua06]) and the fact that $v(t)=0$ for $t\in[n+1/2,n+1]$
(see (3.5)), we obtain

$\displaystyle{\mathbb{E}}\left|\int_{n}^{n+1}v(s){\textup{d}}W(s)\right|^{2}$
$\displaystyle={\mathbb{E}}\int_{n}^{n+1/2}|v(s)|_{{\mathbb{R}}^{d}}^{2}{\textup{d}}s$
$\displaystyle\quad+{\mathbb{E}}\int_{n}^{n+1/2}\\!\\!\\!\int_{n}^{n+1/2}\text{Tr}({\mathcal{D}}_{s}v(r){\mathcal{D}}_{r}v(s)){\textup{d}}s{\textup{d}}r$
$\displaystyle\leq{\mathbb{E}}\int_{n}^{n+1/2}|v(s)|_{{\mathbb{R}}^{d}}^{2}{\textup{d}}s$
$\displaystyle\quad+{\mathbb{E}}\int_{n}^{n+1/2}\\!\\!\\!\int_{n}^{n+1/2}|{\mathcal{D}}_{r}v_{n,n+1/2}(s)|_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}^{2}{\textup{d}}s{\textup{d}}r$
$\displaystyle=L_{1}+L_{2}.$ (3.11)

We estimate $L_{1}$ by using (2.6), (3.7), and (3.8):

$\displaystyle{\mathbb{E}}\int_{n}^{n+1/2}|v(s)|_{{\mathbb{R}}^{d}}^{2}{\textup{d}}s$
$\displaystyle\leq\beta^{-1}{\mathbb{E}}\|J_{n,n+1/2}\rho_{n}\|^{2}$
$\displaystyle\leq
C\beta^{-1}\exp\\{\kappa\|w\|^{2}/2\\}\left({\mathbb{E}}\|\rho_{n}\|^{4}\right)^{1/2}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}-\alpha n/2\\}.$ (3.12)

To estimate $L_{2}$, we use the explicit form of ${\mathcal{D}}_{r}v.$ Notice
that, for any $r\in[n,n+1/2]$ and $i=1,\dots,d$,

$\displaystyle{\mathcal{D}}_{r}^{i}v_{n,n+1/2}$
$\displaystyle={\mathcal{D}}_{r}^{i}({\mathcal{A}}_{n,n+1/2}^{*})({\mathcal{M}}_{n,n+1/2}+\beta{\mathbb{I}})^{-1}J_{n,n+1/2}\rho_{n}$
$\displaystyle\quad+{\mathcal{A}}_{n,n+1/2}^{*}({\mathcal{M}}_{n,n+1/2}+\beta{\mathbb{I}})^{-1}$
$\displaystyle\quad\quad\times\Big{(}{\mathcal{D}}_{r}^{i}({\mathcal{A}}_{n,n+1/2}){\mathcal{A}}_{n,n+1/2}^{*}+{\mathcal{A}}_{n,n+1/2}{\mathcal{D}}_{r}^{i}({\mathcal{A}}_{n,n+1/2}^{*})\Big{)}$
$\displaystyle\quad\quad\times({\mathcal{M}}_{n,n+1/2}+\beta{\mathbb{I}})^{-1}J_{n,n+1/2}\rho_{n}$
$\displaystyle\quad+{\mathcal{A}}_{n,n+1/2}^{*}({\mathcal{M}}_{n,n+1/2}+\beta{\mathbb{I}})^{-1}{\mathcal{D}}_{r}^{i}(J_{n,n+1/2})\rho_{n}.$

By inequalities (2.8)-(2.10), we have

$\displaystyle\|{\mathcal{D}}_{r}^{i}v_{n,n+1/2}\|_{L^{2}([n,n+1/2];{\mathbb{R}}^{d})}$
$\displaystyle\leq\beta^{-1}\|{\mathcal{D}}_{r}^{i}({\mathcal{A}}_{n,n+1/2})\|_{{\mathcal{L}}(L^{2}([n,n+1/2];{\mathbb{R}}^{d}),\tilde{H})}$
$\displaystyle\quad\quad\times\|J_{n,n+1/2}\rho_{n}\|$
$\displaystyle\quad+2\beta^{-1}\|{\mathcal{D}}_{r}^{i}({\mathcal{A}}_{n,n+1/2}^{*})\|_{{\mathcal{L}}(\tilde{H},L^{2}([n,n+1/2];{\mathbb{R}}^{d}))}$
$\displaystyle\quad\quad\times\|J_{n,n+1/2}\rho_{n}\|$
$\displaystyle\quad+\beta^{-1/2}\|{\mathcal{D}}_{r}^{i}(J_{n,n+1/2})\rho_{n}\|.$

Inequalities (2.6), (2.11)-(2.13), and (3.7), imply that

$\displaystyle{\mathbb{E}}\int_{n}^{n+1/2}\\!\\!\\!\int_{n}^{n+1/2}|{\mathcal{D}}_{r}v_{n,n+1/2}(s)|_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}^{2}{\textup{d}}s{\textup{d}}r$
$\displaystyle\leq
C\beta^{-2}\exp\\{\kappa\|w\|^{2}/2\\}\left({\mathbb{E}}\|\rho_{n}\|^{4}\right)^{1/2}$
$\displaystyle\leq C\exp\\{\kappa\|w\|^{2}-n\alpha/2\\}.$ (3.13)

Combining estimates (3.11)-(3.13) and using the fact that $\kappa>0$ and
$\alpha>0$ are arbitrary, we obtain the desired result. ∎

Finally, we will use a growth estimate for the Feynman–Kac semigroup
$\tilde{\mathfrak{P}}^{\tilde{V}}_{t}$. From the first growth esimate in (1.3)
for the semigroup ${\mathfrak{P}}^{V}_{t}$ it follows that there are numbers
$R_{0}>0$, $\gamma\in(0,\gamma_{0})$, and $m\geq 1$ such that

$\tilde{\mathfrak{P}}^{\tilde{V}}_{t}{\mathbf{1}}(w)\leq
C\,{\mathfrak{w}}_{m}({\cal
K}w)\|{\mathfrak{P}}^{V}_{t}{{\mathbf{1}}}\|_{R_{0}}\quad\text{for any
$w\in\tilde{H}$ and $t\geq 0$}.$ (3.14)

Now we are in a position to prove Proposition 3.2.

###### Proof of Proposition 3.2.

Replacing $V$ by $V-\inf_{v\in H}V(v),$ without loss of generality, we can
assume that $V\geq 0$. Let $v$ be the process defined by (3.5), let $\kappa$
and $\alpha$ be positive numbers (to be chosen later), and let the number
$\beta=\beta(\kappa,\alpha)>0$ be such that inequalities (3.6) and (3.10)
hold. Furthermore, let the positive numbers $R_{0}$ and $m$ be such that
inequality (3.14) holds. Then the computations in (3.3) are rigorously
justified, and we need to estimate the terms $I_{1}$, $I_{2}$, and $I_{3}$.

Step 1: Estimate for $I_{1}$. We write the Skorokhod integral in the term
$I_{1}$ (see (3.4)) as follows

$\int_{0}^{t}v(s){\textup{d}}W(s)=\sum_{n=1}^{\lfloor
t\rfloor}\int_{n-1}^{n}v(s){\textup{d}}W(s)+\int_{\lfloor
t\rfloor}^{t}v(s){\textup{d}}W(s),$

where $\lfloor t\rfloor$ is the largest number less than or equal to $t$ and
the sum in the right-hand side is replaced by zero if $t<1$. Since $v(s)$ is
${\mathcal{F}}_{n}$-measurable for $s\in[n-1,n]$, the Skorokhod integral
$\int_{n-1}^{n}v(s){\textup{d}}W(s)$ is also ${\mathcal{F}}_{n}$-measurable.
Hence, using the Markov property, we obtain

$\displaystyle I_{1,n}=$
$\displaystyle\,\,{\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{n-1}^{n}v(s){\textup{d}}W(s)\right]$
$\displaystyle=$
$\displaystyle\,\,{\mathbb{E}}_{w}\left[{\mathbb{E}}_{w}\left(\Xi_{t}\tilde{\psi}(w_{t})\int_{n-1}^{n}v(s){\textup{d}}W(s)\Big{|}{\mathcal{F}}_{n}\right)\right]$
$\displaystyle=$
$\displaystyle\,\,{\mathbb{E}}_{w}\left[\Xi_{n}\int_{n-1}^{n}v(s){\textup{d}}W(s)\,{\mathbb{E}}_{w}\left(\exp\left\\{\int_{n}^{t}\tilde{V}(w_{s}){\textup{d}}s\right\\}\tilde{\psi}(w_{t})\Big{|}{\mathcal{F}}_{n}\right)\right]$
$\displaystyle=$
$\displaystyle\,\,{\mathbb{E}}_{w}\left[\Xi_{n}\int_{n-1}^{n}v(s){\textup{d}}W(s)\,\left(\tilde{\mathfrak{P}}^{\tilde{V}}_{t-n}\tilde{\psi}\right)(w_{n})\right]$

for any $1\leq n\leq\lfloor t\rfloor$. Using inequalities (1.2), (3.10),
(3.14), the assumption that $V\geq 0$, and the Cauchy–Schwarz inequality, we
see that

$\displaystyle I_{1,n}$ $\displaystyle\leq
C\|\psi\|_{\infty}e^{\|V\|_{\infty}n}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}{\mathbb{E}}_{w}\left[{\mathfrak{w}}_{m}(u_{n})\Big{|}\int_{n-1}^{n}v(s){\textup{d}}W(s)\Big{|}\right]$
$\displaystyle\leq
C\|\psi\|_{\infty}e^{\|V\|_{\infty}n}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}\left({\mathbb{E}}_{u}{\mathfrak{w}}_{m}^{2}(u_{n})\right)^{1/2}\left({\mathbb{E}}_{w}\Big{|}\int_{n-1}^{n}v(s){\textup{d}}W(s)\Big{|}^{2}\right)^{1/2}$
$\displaystyle\leq
C\|\psi\|_{\infty}e^{\|V\|_{\infty}n}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}{\mathfrak{w}}_{m}(u)\exp\\{(\kappa\|w\|^{2}-\alpha
n)/2\\},$

where $u={\cal K}w$ and $u_{s}={\cal K}w_{s}$. Next, using (3.10) and $V\geq
0$, we get

$\displaystyle I_{1,\lfloor t\rfloor+1}$
$\displaystyle={\mathbb{E}}_{w}\left[\Xi_{t}\tilde{\psi}(w_{t})\int_{\lfloor
t\rfloor}^{t}v(s){\textup{d}}W(s)\right]$
$\displaystyle\leq\|\psi\|_{\infty}e^{\|V\|_{\infty}t}{\mathbb{E}}_{w}\left|\int_{\lfloor
t\rfloor}^{t}v(s){\textup{d}}W(s)\right|$ $\displaystyle\leq
C\|\psi\|_{\infty}e^{\|V\|_{\infty}t}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}\exp\\{(\kappa\|w\|^{2}-\alpha\lfloor
t\rfloor)/2\\}.$

Combining the estimates for $I_{1,n}$ and $I_{1,\lfloor t\rfloor+1}$, we
arrive at

$\displaystyle I_{1}\leq
C\|\psi\|_{\infty}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}\exp\\{\kappa\|w\|^{2}\\}\sum_{n=1}^{{}^{\lfloor
t\rfloor}}\exp\\{(\|V\|_{\infty}-\alpha/2)n\\}.$

Step 2: Estimate for $I_{2}$. We first write

$\displaystyle I_{2}$ $\displaystyle={\mathbb{E}}_{w}\left[\int_{0}^{\lfloor
t\rfloor}\Xi_{t}\tilde{\psi}(w_{t})\nabla\tilde{V}(w_{s})\rho_{s}{\textup{d}}s\right]+{\mathbb{E}}_{w}\left[\int_{\lfloor
t\rfloor}^{t}\Xi_{t}\tilde{\psi}(w_{t})\nabla\tilde{V}(w_{s})\rho_{s}{\textup{d}}s\right]$
$\displaystyle=I_{2,1}+I_{2,2}.$

Let $\lceil s\rceil$ be the smallest integer greater than or equal to $s$.
Then $\rho(s)$ is ${\mathcal{F}}_{\lceil s\rceil}$-measurable, and using the
Markov property, we obtain

$\displaystyle I_{2,1}$ $\displaystyle={\mathbb{E}}_{w}\left[\int_{0}^{\lfloor
t\rfloor}{\mathbb{E}}_{w}\left(\Xi_{t}\tilde{\psi}(w_{t})\nabla\tilde{V}(w_{s})\rho_{s}\big{|}{\mathcal{F}}_{\lceil
s\rceil}\right){\textup{d}}s\right]$
$\displaystyle={\mathbb{E}}_{w}\left[\int_{0}^{\lfloor t\rfloor}\Xi_{\lceil
s\rceil}\nabla\tilde{V}(w_{s})\rho_{s}\
{\mathbb{E}}_{w}\left(\exp\left\\{\int_{\lceil
s\rceil}^{t}\tilde{V}(w_{r}){\textup{d}}r\right\\}\tilde{\psi}(w_{t})\Big{|}{\mathcal{F}}_{\lceil
s\rceil}\right){\textup{d}}s\right]$
$\displaystyle={\mathbb{E}}_{w}\left[\int_{0}^{\lfloor t\rfloor}\Xi_{\lceil
s\rceil}\nabla\tilde{V}(w_{s})\rho_{s}\left(\tilde{\mathfrak{P}}^{\tilde{V}}_{t-\lceil
s\rceil}\tilde{\psi}\right)(w_{\lceil s\rceil}){\textup{d}}s\right].$

Then inequalities (1.2), (3.6), (3.14), the assumption that $V\geq 0$, and the
Cauchy–Schwarz inequality imply that

$\displaystyle I_{2,1}$ $\displaystyle\leq C\|\psi\|_{\infty}\|\nabla
V\|_{\infty}\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\int_{0}^{\lfloor
t\rfloor}e^{\|V\|_{\infty}\lceil
s\rceil}{\mathbb{E}}_{w}\left[{\mathfrak{w}}_{m}(u_{\lceil
s\rceil})\|\rho_{s}\|\right]{\textup{d}}s$ $\displaystyle\leq
C\|\psi\|_{\infty}\|\nabla
V\|_{\infty}\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\int_{0}^{\lfloor
t\rfloor}e^{\|V\|_{\infty}\lceil
s\rceil}\left({\mathbb{E}}_{u}{\mathfrak{w}}_{m}^{2}(u_{\lceil
s\rceil})\right)^{1/2}\left({\mathbb{E}}_{w}\|\rho_{s}\|^{2}\right)^{1/2}{\textup{d}}s$
$\displaystyle\leq C\|\psi\|_{\infty}\|\nabla
V\|_{\infty}\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}{\mathfrak{w}}_{m}(u)\exp\\{\kappa\|w\|^{2}/4\\}$
$\displaystyle\quad\quad\quad\quad\times\int_{0}^{\lfloor
t\rfloor}\exp\\{\|V\|_{\infty}\lceil s\rceil-\alpha s/4\\}{\textup{d}}s.$

To estimate $I_{2,2}$, we use (3.6) and $V\geq 0$:

$\displaystyle I_{2,2}$
$\displaystyle\leq\|\psi\|_{\infty}e^{\|V\|_{\infty}t}\|\nabla
V\|_{\infty}{\mathbb{E}}_{w}\int_{\lfloor
t\rfloor}^{t}\|\rho_{s}\|{\textup{d}}s$ $\displaystyle\leq
C\|\psi\|_{\infty}e^{\|V\|_{\infty}t}\|\nabla
V\|_{\infty}\|{\mathfrak{P}}_{t}^{V}{\mathbf{1}}\|_{R_{0}}\exp\\{(\kappa\|w\|^{2}-\alpha
t)/4\\}.$

Thus

$\displaystyle I_{2}$ $\displaystyle\leq C\|\psi\|_{\infty}\|\nabla
V\|_{\infty}\|{\mathfrak{P}}_{t}^{V}\textbf{1}\|_{R_{0}}\exp\\{\kappa\|w\|^{2}\\}$
$\displaystyle\quad\quad\quad\quad\times\left(\exp\\{-\alpha
t)/4\\}+\int_{0}^{\lfloor t\rfloor}\exp\\{\|V\|_{\infty}\lceil s\rceil-\alpha
s/4\\}{\textup{d}}s\right).$

Step 3: Estimate for $I_{3}$. By (3.6), we have

$|I_{3}|\leq
C\|\nabla\psi\|_{\infty}e^{\|V\|_{\infty}t}{\mathbb{E}}\|\rho_{t}\|\leq\|\nabla\psi\|_{\infty}e^{\|V\|_{\infty}t}\exp\\{(\kappa\|w\|^{2}-t\alpha)/4\\}.$

Choosing $\alpha\geq 4\|V\|_{\infty}-\log a$ and combining the above estimates
of the terms $I_{i}$, $i=1,2,3$ with (3.3), we complete the proof of
Proposition 3.2. ∎

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