# Large deviations for 2D stochastic Navier-Stokes Equations driven by a
periodic force and a degenerate noise

Rongchang Liu  Department of Mathematics
University of Arizona
Tucson, AZ 85721, USA<EMAIL_ADDRESS>and  Kening Lu  School of
Mathematics
Sichuan University
Chengdu, Sichuan 610064, PR China<EMAIL_ADDRESS>
###### Abstract.

We consider the incompressible 2D Navier-Stokes equations on the torus, driven
by a deterministic time periodic force and a noise that is white in time and
degenerate in Fourier space. The main result is twofold.

Firstly, we establish a Ruelle-Perron-Frobenius type theorem for the time
inhomogeneous Feynman-Kac evolution operators with regular potentials
associated with the stochastic Navier-Stokes system. The theorem characterizes
asymptotic behaviors of the Feynman-Kac operators in terms of the periodic
family of principal eigenvalues and corresponding unique eigenvectors. The
proof involves a time inhomogeneous version of Ruelle’s lower bound technique.

Secondly, utilizing this Ruelle-Perron-Frobenius type theorem and a Kifer’s
criterion, we establish a Donsker-Varadhan type large deviation principle with
a nontrivial good rate function for the occupation measures of the time
inhomogeneous solution processes.

###### Key words and phrases:

Navier-Stokes Equations, periodic Ruelle-Perron-Frobenius theorem, Donsker-
Varadhan large deviation principle, degenerate noise, time inhomogeneous
Markov processes.

###### 1991 Mathematics Subject Classification:

Primary: 60H15, 60F10, 76D05, 37A05

###### Contents

  1. 1 Introduction
  2. 2 Settings and main results
    1. 2.1 Time symbols and Feynman-Kac operators
    2. 2.2 Main theorems
  3. 3 A Ruelle-Perron-Frobenius type theorem
    1. 3.1 Existence of eigen-triple
    2. 3.2 Forward and pullback stability for discrete time
      1. 3.2.1 Forward stability in $L^{1}$ norm
      2. 3.2.2 Proof of Theorem 3.3
    3. 3.3 Uniqueness and regularity of the eigen-triple
    4. 3.4 Proof of Theorem 2.1
  4. 4 A Donsker-Varadhan type large deviation principle
    1. 4.1 Proof of Theorem 2.2
      1. 4.1.1 Proof of Theorem 2.2
      2. 4.1.2 Proof of Theorem 4.1
    2. 4.2 Proof of auxiliary lemmas
      1. 4.2.1 Time continuity and Feller property of $\mathcal{T}_{t}^{V}$
      2. 4.2.2 Proof of Lemma 4.3
  5. A A Kifer type criterion
  6. B A priori estimates of solutions
  7. C Growth condition, uniform irreducibility and uniform Feller property
    1. C.1 The Growth Condition
    2. C.2 Uniform Irreducibility
    3. C.3 Uniform Feller Property
      1. C.3.1 Malliavin calculus preliminaries
      2. C.3.2 Proof of Proposition C.5

##  1\. Introduction

The 2D Navier-Stokes equations subject to a time periodic force has been
extensively studied over the years, dating back to Serrin [36], Yudovich [38]
and many others, see Galdi and Kyed [16] for recent progress. It was proved by
Da Prato and Debussche [10] that the periodically forced 2D Navier-Stokes
system with a non-degenerate noise has an exponentially mixing periodic
invariant measure. Furthermore, it was proved recently [28] that when the
noise is degenerate, the strong law of large numbers and central limit theorem
hold. In particular, the central limit theorem describes the small deviations
from the periodic invariant measure.

The aim of the current paper is to investigate the probabilities of large
deviations from the periodic invariant measure, for the periodically forced 2D
Navier-Stokes equations with a degenerate white in time noise. To the best of
our knowledge, this work seems to be the first one that establishes Donsker-
Varadhan type large deviation results for the occupation measures of the time
inhomogeneous solution processes solving a given SPDE. In what follows, we
will briefly describe our results (Theorems A and B below), while leave the
technical preliminaries to the next section.

Consider the incompressible 2D Navier-Stokes equations on the torus
$\mathbb{T}^{2}:=\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$ in the following vorticity
form

(1.1) $\displaystyle
dw(t,x)+B(\mathcal{K}w,w)(t,x)dt=\nu\mathrm{\mathrm{\Delta}}w(t,x)dt+f(t,x)dt+\sum_{i=1}^{d}g_{i}dW_{i}(t),\quad
t>s,\quad w(s)=w_{0},$

for $s\in\mathbb{R},x\in\mathbb{T}^{2}$ and some integer $d\geq 1$. Here $w$
is the vorticity field, $\mathcal{K}$ is the Biot-Savart integral operator and
$B(\mathcal{K}w,w)=(\mathcal{K}w)\cdot\nabla w$ is the nonlinearity. The phase
space is chosen as
$H:=\left\\{w\in\mathrm{L}^{2}\left(\mathbb{T}^{2},\mathbb{R}\right):\int_{\mathbb{T}^{2}}wdx=0\right\\}$
whose norm is denoted by $\|\cdot\|$ and the initial $w_{0}\in H$. We also
denote $H_{s}=H^{s}\cap H$ and the corresponding norm as
$\left\|\cdot\right\|_{s}$ where $H^{s}$ is the usual Sobolev space of order
$s$. The deterministic force $f$ is $2\pi$-periodic in $t$. The noise
$W=(W_{1},W_{2},\cdots,W_{d})$ is a two-sided $\mathbb{R}^{d}$-valued standard
Wiener process over the sample space $(\Omega,\mathcal{F},\mathbf{P})$ where
$\mathbf{P}$ is the Wiener measure, and the noise coefficients $\\{g_{i}\\}$
are smooth elements of $H$.

We denote the solution to (1.1) by $\Phi_{s,t}(w_{0})$. To introduce the
Feynman-Kac operators in our time inhomogeneous setting, we utilize the time
symbols formulation used in deterministic systems, see Chepyzhov and Vishik
[8] for example. To be precise, we consider a family of Navier-Stokes systems
indexed by time symbols $h\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$,

(1.2) $\displaystyle
dw(t,x)+B(\mathcal{K}w,w)(t,x)dt=\nu\mathrm{\mathrm{\Delta}}w(t,x)dt+f(\beta_{t}h,x)dt+\sum_{i=1}^{d}g_{i}dW_{i}(t),\,t>s,\,w(s)=w_{0},$

whose solution is denoted by $\Phi_{s,t,h}(w_{0})$. Here $\beta_{t}h=h+t$ is
the circle rotation on $\mathbb{T}$. Note that (1.2) when $h=0$ is the
original equation (1.1).

Now for every potential $V\in C_{b}(H\times\mathbb{T})$, the associated
Feynman-Kac operator $P_{s,t,h}^{V}:C_{b}(H)\to C_{b}(H)$ corresponding to
(1.2) with time symbol $h\in\mathbb{T}$ is given by

(1.3) $\displaystyle
P_{s,t,h}^{V}\phi(w)=\mathbf{E}\left[\exp\left(\int_{s}^{t}V(\Phi_{s,r,h}(w),\beta_{r}h)dr\right)\phi(\Phi_{s,t,h}(w))\right],\,\phi\in
C_{b}(H),w\in H.$

It naturally induces by duality an action $P_{s,t,h}^{V*}$ on the space
$\mathcal{M}_{+}(H)$ of non-negative finite Borel measures on $H$, which can
be regarded as an averaged transfer operator.

Since the paper is working with a degenerate noise, we recall the Lie bracket
condition from Hairer and Mattingly [21]. Define the set $A_{\infty}$ by
setting

(1.4) $\displaystyle A_{1}=\\{g_{l}:1\leq l\leq
d\\},A_{k+1}=A_{k}\cup\\{\widetilde{B}(u,v):u,v\in
A_{k}\\},A_{\infty}=\overline{\mathrm{span}(\cup_{k\geq 1}A_{k})},$

where $\widetilde{B}(u,w)=-B(\mathcal{K}u,w)-B(\mathcal{K}w,u)$ is the
symmetrized nonlinear term. The assumptions for our main results will be the
Hörmander condition $A_{\infty}=H$ and regularity conditions on the force $f$.

Ruelle-Perron-Frobenius type theorem for the Feynman-Kac operators

In the study of an infinite one dimensional lattice gas model, Ruelle [35]
proved his Perron-Frobenius type theorem for the corresponding transfer
operator with Hölder regular potentials. This theorem is now known as Ruelle-
Perron-Frobenius (RPF) theorem and has been widely used and extended in the
study of dynamical systems and thermodynamic formalism, see Baladi [3], Bowen
[6], Climenhaga and Pesin [7] and references therein. Ruelle’s ideas in [35]
were then used by Kuksin and Shirikyan [25] to show unique ergodicity and
asymptotic stability of Markov operators for a class of dissipative PDEs.
Later the same ideas were applied by Jakšić et al. [22] to the discrete time
Feynman-Kac semigroups associated with the models considered in [25] for which
an RPF type theorem was proved in a compact setting. An extension to the non-
compact setting was obtained by the same authors in [23], which was further
extended to continuous time by Martirosyan and Nersesyan [29]. All the
existing results work with time homogeneous Feynman-Kac semigroups.

Our Theorem A below gives a version of the RPF type theorem for the time
inhomogeneous Feynman-Kac evolution operators of (1.1), which is effective in
establishing LDP for the time inhomogeneous solution processes. Here the
effectiveness means that the eigen-elements are continuous on the time symbols
which enables us to establish the LDP for the original system (1.1) from
Theorem A. Below $\mathcal{P}(H)$ is the space of probability measures on $H$.

######  Theorem A (RPF Type Theorem).

Assume $A_{\infty}=H$ and $f\in C(\mathbb{R},H_{2})$ is $2\pi$-periodic and
Lipschitz. For each potential $V\in C_{b}^{1}(H\times\mathbb{T})$ there exists
a unique family of eigen-triple of the Feynman-Kac operators (1.3) associated
with the Navier-Stokes family (1.2),

$\displaystyle(\Gamma^{V},\lambda^{V},F^{V})\in
C(\mathbb{T},\mathcal{P}(H))\times C(\mathbb{T},\mathbb{R})\times
C(H\times\mathbb{T}),$

with $F^{V}$ strictly positive and normalized by $\Gamma^{V}$ such that the
following properties hold:

  1. (1)

Eigenmeasure/eigenfunction property with positive eigenvalue
$\lambda_{t,h,V}=\exp\left(\int_{0}^{t}\lambda^{V}(\beta_{r}h)dr\right)$,

(1.5)
$\displaystyle\begin{split}&P_{0,t,h}^{V*}\Gamma^{V}(h)=\lambda_{t,h,V}\Gamma^{V}(\beta_{t}h),\qquad\text{(eigenmeasure
property)}\\\
&P_{0,t,h}^{V}F^{V}(\cdot,\beta_{t}h)(w)=\lambda_{t,h,V}F^{V}(\cdot,h),\quad\text{(eigenfunction
property)}\end{split}$

for all $h\in\mathbb{T}$ and $t\geq 0$.

  2. (2)

Forward and pullback stability: for any $h\in\mathbb{T}$, $\phi\in C_{b}(H)$
and $w\in H$,

(1.6)
$\displaystyle\begin{split}&\lim_{t\to\infty}\left|\lambda^{-1}_{t,h,V}P_{0,t,h}^{V}\phi(w)-\left\langle\Gamma^{V}(\beta_{t}h),\phi\right\rangle
F^{V}(w,h)\right|=0,\\\
&\lim_{t\to\infty}\left|\lambda^{-1}_{t,\beta_{-t}h,V}P_{0,t,\beta_{-t}h}^{V}\phi(w)-\left\langle\Gamma^{V}(h),\phi\right\rangle
F^{V}(w,\beta_{-t}h)\right|=0.\end{split}$

Theorem A is a simplified version of Theorem 2.1, which will be proved in
section 3. Note that when $h=0$ the theorem gives the RPF type theorem for the
Feynman-Kac operators associated with (1.1). Theorem A is proved by first
showing a weak version (without continuous dependence on time symbols) for the
discrete time restrictions and then passing to the whole continuous time.

A major step in proving the weak version is to show the forward $L^{1}$
stability after one obtains the existence of the eigen-elements. In [25, 23],
this was proved by combining the uniform Feller property and irreducibility
with Ruelle’s lower bound technique, which relies on an $L^{1}$ non-expansive
property. Due to the time inhomogeneity, such non-expansiveness is not
available in the current setting. The approach we take in the proof provides a
way to tailor the core of Ruelle’s lower bound technique in this situation, to
obtain the desired forward $L^{1}$ stability. The impact of time inhomogeneity
is also reflected throughout the process.

The passage from the weak version to the whole continuous time is accomplished
through the eigen-properties and uniqueness. Then we establish the continuity
of eigen-triple and the representation of the eigenvalues through a common
$\lambda^{V}$. The continuity of the eigenmeasure (and eigenvalue) is obtained
by controlling the lower bound of the evaluation of eigenmeasure along the
eigenfunction. The continuity of the eigenfunction is proved by combining the
eigenfunction property (1.5) and the time continuity of the Feynman-Kac
family, as well as the dynamical structure of the time inhomogeneity.

Donsker-Varadhan type large deviation principle

For time homogeneous Markov processes, asymptotic statistical behaviors can be
described by the strong law of large numbers and central limit theorem when
there is a mixing invariant measure, see Kuksin and Shirikyan [26] for
example. In particular, the central limit theorem characterizes fluctuations
of the occupation measures around the invariant measure. A more detailed
analysis enables one to derive a LDP that quantifies the exponentially
decaying rates of the probabilities of large deviations from the invariant
measure, see for example the pioneering works of Donsker and Varadhan [12,
13].

For time inhomogeneous Markov processes, the asymptotic statistical profile
depends on the time inhomogeneity. It was proved recently [28] that the
statistical dynamics of the time inhomogeneous solution processes for the
Navier-Stokes equations (1.1) is characterized by an exponential mixing
periodic invariant measure $\Gamma\in C(\mathbb{T},\mathcal{P}(H))$. In
addition, one has the following strong law of large numbers and central limit
theorem:

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\big{(}\phi(\Phi_{0,r})-\left\langle\Gamma(\beta_{r}0),\phi\right\rangle\big{)}dr\to
0,\,\text{ almost surely, }$
$\displaystyle\lim_{t\to\infty}\frac{1}{\sqrt{t}}\int_{0}^{t}\big{(}\phi(\Phi_{0,r})-\left\langle\Gamma(\beta_{r}0),\phi\right\rangle\big{)}dr=N(0,\sigma^{2}),\,\text{
in distribution, }$

where recall that $\beta_{r}$ is the circle rotation (1.2) and $\Phi_{0,r}$ is
the solution to (1.1). Here $N(0,\sigma^{2})$ is a centered normal random
variable with variance $\sigma^{2}$. The central limit theorem describes the
distribution of the fluctuations around the periodic invariant measure.

Our Theorem B below gives quantitative information on the probabilities of
large deviations from the periodic invariant measure. Denote the occupation
measure by

$\displaystyle\zeta_{t}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\delta_{(\Phi_{0,r},\beta_{r}0)}dr,$

which is a family of random probability measures on $H\times\mathbb{T}$. In
the following theorem $\mathcal{P}(H\times\mathbb{T})$ is the space of
probability measures on $H\times\mathbb{T}$ with the topology of weak
convergence.

######  Theorem B (Large Deviation Principle).

Assume $A_{\infty}=H$ and $f\in C(\mathbb{R},H_{2})$ is $2\pi$-periodic and
Lipschitz. Then there is a non-trivial good rate function
$I:\mathcal{P}(H\times\mathbb{T})\to[0,\infty]$ such that for every initial
vorticity $w\in H$, we have

$\displaystyle-\inf_{\mu\in\mathring{B}}I(\mu)\leq\liminf_{t\to\infty}\frac{1}{t}\log\mathbf{P}_{w}\\{\zeta_{t}\in
B\\}\leq\limsup_{t\to\infty}\frac{1}{t}\log\mathbf{P}_{w}\\{\zeta_{t}\in
B\\}\leq-\inf_{\mu\in\overline{B}}I(\mu),$

for any Borel subset $B$ in $\mathcal{P}(H\times\mathbb{T})$, with
$\mathring{B}$ and $\overline{B}$ its interior and closure.

Theorem B is a simplified version of Theorem 2.2, which will be proved in
section 4. It immediately implies probability bounds for large deviations from
the unique periodic invariant measure $\Gamma$. Indeed, for any Borel set
$E\subset\mathbb{R}$ and observable function $\phi\in C_{b}(H)$, define the
Borel subset $B$ of $\mathcal{P}(H\times\mathbb{T})$,

$B:=\left\\{\mu\in\mathcal{P}(H\times\mathbb{T}):\left\langle\mu,\phi_{\Gamma}\right\rangle\in
E\right\\},$

where $\phi_{\Gamma}(w,h)=\phi(w)-\left\langle\Gamma(h),\phi\right\rangle$ and
$\left\langle\mu,\phi\right\rangle$ denotes the integral of a function $\phi$
with respect to a measure $\mu$. Note that

$\displaystyle\zeta_{t}\in B\text{ is equivalent to
}\left\langle\zeta_{t},\phi_{\Gamma}\right\rangle=\frac{1}{t}\int_{0}^{t}\big{(}\phi(\Phi_{0,r})-\left\langle\Gamma(\beta_{r}0),\phi\right\rangle\big{)}dr\in
E.$

Thus by Theorem B one obtains when $t\to\infty$,

$\displaystyle
e^{-tL^{-}}\lesssim\mathbf{P}_{w}\left\\{\frac{1}{t}\int_{0}^{t}\big{(}\phi(\Phi_{0,r})-\left\langle\Gamma(\beta_{r}0),\phi\right\rangle\big{)}dr\in
E\right\\}\lesssim e^{-tL^{+}},$

where $L^{\pm}=L^{\pm}(\phi_{\Gamma},E)\geq 0$ are constants determined by the
rate function.

The theorem is proved through a combination of the RPF type Theorem A and a
Kifer’s criterion established by Jakšić et al. [23]. It is known that
existence of the pressure function and exponential tightness imply the large
deviation upper bound with its Legendre transform as the rate function. Such
existence in the time homogeneous case is a direct consequence of the
stability of Feynman-Kac semigroups but more effort is needed here. The
existence here follows by a combination of the forward stability (1.6),
integral representation for $\lambda_{t,h,V}$ in (1.5), and the following
Birkhoff ergodic theorem for circle rotation with unique invariant Lebesgue
measure $m$,

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\lambda^{V}(\beta_{r}0)dr=\int_{\mathbb{T}}\lambda^{V}(h)m(dh),$

which is ensured by the continuity of $\lambda^{V}$ established in Theorem A.
We emphasize that if there is no continuity then Birkhoff’s ergodic theorem is
not necessarily true for the time symbol $0$ and therefore the existence of
the pressure function for the given system (1.1) would become an issue.

The large deviation lower bound requires additionally the uniqueness of
equilibrium states for a rich class of regular potentials. This is usually
done by connecting the equilibrium state with the unique invariant measure of
an auxiliary Markov semigroup obtained from the Feynman-Kac semigroup by
Doob’s $h$-transform. We follow this approach but additional issues arise due
to time inhomogeneity. Firstly, the Doob’s $h$-transform of the inhomogeneous
Feynman-Kac family is still inhomogeneous. To establish the uniqueness of
invariant measure for its homogenization $\mathcal{T}_{t}^{V}$, we make a
correspondence between their invariant measures, so that we can apply the
uniqueness of eigen-triple in Theorem A. Secondly, to utilize the Donsker-
Varadhan functional of $\mathcal{T}_{t}^{V}$ to establish an effective
connection between equilibrium states and invariant measures, we show that the
domain of the infinitesimal generator of $\mathcal{T}_{t}^{V}$ is rich enough
to be determining. This involves a verification of the Feller property of
$\mathcal{T}_{t}^{V}$, where the continuity on time symbols of the eigen-
triple plays a crucial role.

We end the introduction with a brief review on the Donsker-Varadhan type large
deviations from a stationary distribution for 2D Navier-Stokes equations. It
is worth mentioning that the first such LDP for SPDEs was given by Gourcy [17,
18]. By assuming that the white in time noise is sufficiently irregular in
space, he established the LDP for occupation measures of the stochastic
Burgers and Navier-Stokes equations, using a general result of Wu [37] that
requires the strong Feller property. Utilizing the approach that combines the
asymptotic behavior of Feynman-Kac semigroups with a Kifer type criterion,
Jakšić et al. [22] established the first result on LDP for dissipative SPDEs
when the strong Feller property does not hold, including stochastic Navier-
Stokes equations perturbed by a smooth bounded random kick force. The authors
in [22] extended their work to the case of smooth unbounded random kick forces
in [23], which was extended by Martirosyan and Nersesyan [29, 30] to the
continuous time case, giving a proof of the LDP for the 2D Navier-Stokes
equation with a non-degenerate white in time noise that is smooth in space.
Combining this approach with Malliavin calculus, Nersesyan, Peng and Xu [31]
proved the LDP in the case when there is no deterministic force and the white
noise is degenerate in Fourier space. The LDP for 2D Navier-Stokes equations
with a space-time localised noise was also considered in Peng and Xu [34] by
using the same approach. The solution processes considered by these works are
time homogeneous.

Notations

Let $X$ be a Banach space and $Y$ a polish space.
$\mathbb{T}=\mathbb{R}/(2\pi)\mathbb{Z}$ is the $1$-dimensional torus with
Lebesgue measure $m$ normalized as a probability measure.

  * •

$\mathbb{1}_{A}$ is the indicator function of a set $A$. We use $m_{V},M_{V}$
to denote the infimum and supremum of a bounded function $V$ on its whole
domain.

  * •

We denote $B_{r}^{X}(a)$ the closed ball in $X$ centered at $a$ with radius
$r$. When $a=0$ is the origin, we simply denote the ball by $B_{r}^{X}$. The
open ball is denoted by $\mathring{B}_{r}^{X}(a)$ (or $\mathring{B}_{r}^{X}$
if $a=0$).

  * •

$C_{b}(Y)$ (respectively $B_{b}(Y)$) is the space of bounded continuous
(measurable) functions on $Y$ with the usual supremum norm
$\|\cdot\|_{\infty}$. $C_{b}^{1}(X)$ is the subset of $C_{b}(X)$ that is
Fréchet differentiable with a bounded continuous derivative.

  * •

For any measurable function $\mathfrak{m}\geq 1$ on $X$,
$C_{\mathfrak{m}}(X\times\mathbb{T})$ (respectively
$L_{\mathfrak{m}}^{\infty}(X\times\mathbb{T})$) is the space of continuous
(measruable) functions $f$ on $X\times\mathbb{T}$ such that

$\|f\|_{L_{\mathfrak{m}}^{\infty}}=\sup_{(x,h)\in
X\times\mathbb{T}}\frac{|f(x,h)|}{\mathfrak{m}(x)}<\infty.$

  * •

The space of regular potential functions $C_{b}^{1,0}(X\times\mathbb{T})$ is
the set of functions $f\in C_{b}(X\times\mathbb{T})$ that are continuously
Fréchet differentiable in $x\in X$ and continuous in $h$ such that

$\sup_{h\in\mathbb{T}}\|\nabla f(\cdot,h)\|_{\infty}+\sup_{x\in X,h_{1}\neq
h_{2}}\frac{|f(x,h_{1})-f(x,h_{2})|}{\kappa(|h_{1}-h_{2}|)}<\infty,$

for some $\kappa:[0,\infty)\to[0,\infty)$ that is continuous, increasing with
$\kappa(0)=0$.

  * •

$\mathcal{M}_{+}(Y)$ is the space of non-negative finite Borel measures on $Y$
with the topology of weak convergence. Given any bounded measurable function
$f$ on $Y$ and $\mu\in\mathcal{M}_{+}(Y)$, we write

$\langle\mu,f\rangle=\langle f,\mu\rangle=\int_{Y}f(y)\mu(dy),\quad\text{ and
}|f|_{\mu}=\int_{Y}|f(y)|\mu(dy).$

  * •

$\mathcal{P}(Y)\subset\mathcal{M}_{+}(Y)$ is the space of probability measures
on $Y$. $\mathcal{P}_{\mathfrak{m}}(Y)$ is the set of probability measures
$\mu$ such that $\langle\mu,\mathfrak{m}\rangle<\infty$. For $\eta,M>0$, we
denote

$\mathcal{P}_{\eta,M}(H)=\left\\{\mu\in\mathcal{P}(H):\int_{H}e^{\eta\|w\|^{2}}\mu(dw)\leq
M\right\\}.$

  * •

$H:=\left\\{w\in\mathrm{L}^{2}\left(\mathbb{T}^{2},\mathbb{R}\right):\int_{\mathbb{T}^{2}}wdx=0\right\\}$,
where the norm is denoted by $\left\|\cdot\right\|$ and the inner product is
$\langle\cdot,\cdot\rangle$. $H_{s}=H^{s}\cap H$ and the corresponding norms
$\left\|\cdot\right\|_{s}$ by
$\|w\|_{s}=\left\|\left(-\mathrm{\Delta}\right)^{s/2}w\right\|$.

  * •

For any subset $A\subset H$, we denote $(A,\|\cdot\|)$ the topological space
with subspace topology induced from $H$. And for any $a\in H$, we denote
$B_{R}^{H,A}(a):=B_{R}^{H}(a)\cap A$ (or $B_{R}^{H,A}$ if $a=0$) as the closed
ball in $(A,\|\cdot\|)$.

  * •

Lyapunov functions on $H$: $\mathfrak{m}_{\eta}(w)=e^{\eta\|w\|^{2}}$ and
$\mathfrak{m}_{q}(w)=1+\|w\|^{2q}$, for $w\in H$ and $\eta,q>0$.

  * •

$\lceil a\rceil$ is the smallest integer greater than or equal to $a$.
$\phi^{+}$ (respectively $\phi^{-}$) is the positive (negative) part of a
function.

  * •

$\Phi_{s,t,h}$ (respectively $\Phi_{s,t}=\Phi_{s,t,0}$) is the solution to
equations (2.2) (equations (2.1)). $P_{s,t,h}^{V}$ is the Feynman-Kac operator
(2.5) with potential $V$. $\mathcal{H}_{t}$ is the homogenized process (2.4)
and $\mathcal{P}_{t}^{V}$ is its Feynman-Kac semigroup (2.8).
$P_{s,t,h}=P_{s,t,h}^{0}$ and $\mathcal{P}_{t}=\mathcal{P}_{t}^{0}$ are the
corresponding Markov operators.

##  2\. Settings and main results

Recall from the introduction that the equation under investigation is

(2.1) $\displaystyle
dw(t,x)+B(\mathcal{K}w,w)(t,x)dt=\nu\mathrm{\mathrm{\Delta}}w(t,x)dt+f(t,x)dt+\sum_{i=1}^{d}g_{i}dW_{i}(t),\quad
t>s,\quad w(s)=w_{0},$

for $s\in\mathbb{R},x\in\mathbb{T}^{2}$ and some integer $d\geq 1$.

The standard $d$-dimensional two-sided Brownian motion
$W=(W_{1},W_{2},\cdots,W_{d})$ is obtained as follows. Let $W^{\pm}(t)$ be two
independent standard $d$-dimensional Brownian motion, then define

$\displaystyle W(t):=\left\\{\begin{array}[]{rr}W^{+}(t)\text{ , }t\geq 0,\\\
W^{-}(-t)\text{ , }t<0.\\\ \end{array}\right.$

The sample space is denoted by
$(\mathrm{\mathrm{\Omega}},\mathcal{F},\mathbf{P})$, where
$\mathrm{\Omega}=\left\\{\omega\in
C(\mathbb{R},\mathbb{R}^{d}):\omega(0)=0\right\\}$ endowed with the compact
open topology, $\mathcal{F}$ is the Borel $\sigma$-algebra and $\mathbf{P}$ is
the Wiener measure associated with the Brownian motion $W$. Denote by
$\mathcal{F}_{t}:=\sigma(W(u)-W(v):-\infty<u,v\leq t)$ the filtration of
$\sigma$-algebras generated by $W$. The coefficients of the noise
$\left\\{g_{i}\right\\}\subset\bigcap_{s>0}H_{s}$.

For the deterministic force, we assume that $f\in C_{b}(\mathbb{R},H_{2})$ is
$2\pi$-periodic and Lipschitz. The well-posedness of (2.1) under the above
conditions is classical [26]. For every $s\in\mathbb{R}$ and $w_{0}\in H$, the
solution belongs to $C([s,\infty);H)\cap
L_{\mathrm{loc}}^{2}((s,\infty);H_{3})$ almost surely and it generates a
random dynamical system. We denote the solution at time $t\geq s$ by
$\Phi_{s,t}$.

###  2.1. Time symbols and Feynman-Kac operators

Denote by $\beta_{t}$ the circle rotation on the torus
$\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$, i.e.,

$\displaystyle\beta_{t}h=t+h\mod 2\pi,h\in\mathbb{T},t\in\mathbb{R}.$

We consider a family of Navier-Stokes equations indexed by time symbols
$h\in\mathbb{T}$ which is obtained from (2.1) by replacing $f(t,x)$ with
$f(\beta_{t}h,x)$,

(2.2) $\displaystyle
dw(t,x)+B(\mathcal{K}w,w)(t,x)dt=\nu\mathrm{\mathrm{\Delta}}w(t,x)dt+f(\beta_{t}h,x)dt+\sum_{i=1}^{d}g_{i}dW_{i}(t),\,t>s,\,w(s)=w_{0},$

with corresponding solution $\Phi_{s,t,h}(w_{0})$. Note that the symbol $h=0$
corresponds to the original equation (2.1). For any $s\in\mathbb{R},t\geq
0,(w,h)\in H\times\mathbb{T}$, It follows from the uniqueness of solution that

(2.3)
$\displaystyle\Phi_{s,s+t,h}(\omega,w)=\Phi_{0,t,\beta_{s}h}(\theta_{s}\omega,w),\,\mathbf{P}\text{-}\mathrm{a.s.},$

where $\theta$ is the Wiener shift. The coupled process

(2.4) $\displaystyle\mathcal{H}_{t}(w,h)=(\Phi_{0,t,h}(w),\beta_{t}h)$

is a homogeneous Markov process on $H\times\mathbb{T}$.

For every symbol $h\in\mathbb{T}$ and potential $V\in
C_{b}(H\times\mathbb{T})$, the associated Feynman-Kac operator
$P_{s,t,h}^{V}:C_{b}(H)\to C_{b}(H)$ corresponding to (2.2) is defined by

(2.5) $\displaystyle
P_{s,t,h}^{V}\phi(w)=\mathbf{E}\left[\exp\left(\int_{s}^{t}V(\Phi_{s,r,h}(w),\beta_{r}h)dr\right)\phi(\Phi_{s,t,h}(w))\right],\,\phi\in
C_{b}(H),w\in H.$

It naturally induces by duality an action $P_{s,t,h}^{V*}$ on the space
$\mathcal{M}_{+}(H)$ of non-negative finite Borel measures on $H$,

$\displaystyle\left\langle
P_{s,t,h}^{V*}\mu,\phi\right\rangle=\left\langle\mu,P_{s,t,h}^{V}\phi\right\rangle,\,\mu\in\mathcal{M}_{+}(H),\phi\in
C_{b}(H).$

It also satisfies the following evolution relation by the Markov property

$\displaystyle P_{u,t,h}^{V}=P_{u,s,h}^{V}P_{s,t,h}^{V},\,\text{ for all
}u\leq s\leq t,\,h\in\mathbb{T}.$

The translation relation (2.3) of solutions yields the following translation
identity of $P_{s,t,h}^{V}$,

(2.6) $\displaystyle
P_{s,s+t,h}^{V}=P_{0,t,\beta_{s}h}^{V},\,s\in\mathbb{R},t\geq
0,h\in\mathbb{T}.$

Combining this with the evolution relation we obtain the following cocycle
property

(2.7) $\displaystyle
P_{0,s+t,h}^{V}=P_{0,s,h}^{V}P_{0,t,\beta_{s}h}^{V},\,s,t\geq
0,h\in\mathbb{T}.$

It is also convenient to use the homogenized Feynman-Kac semigroup
$\mathcal{P}_{t}^{V}$ acting on $C_{b}(H\times\mathbb{T})$ as

(2.8)
$\displaystyle\mathcal{P}_{t}^{V}\phi(w,h)=P_{0,t,h}^{V}\phi(\cdot,\beta_{t}h)(w),\,t\geq
0,(w,h)\in H\times\mathbb{T},\phi\in C_{b}(H\times\mathbb{T}).$

The Feynman-Kac operators with vanishing potential are the usual Markov
operators, which we denote by $P_{s,t,h}:=P^{0}_{s,t,h}$, as well as
$\mathcal{P}_{t}:=\mathcal{P}_{t}^{0}$.

###  2.2. Main theorems

The following two theorems are our main results under the standing assumption:

(2.9) $\displaystyle f\in C(\mathbb{R},H_{2})\text{ is }2\pi\text{-periodic
and Lipschitz};g_{i}\in H_{\infty},\forall 1\leq i\leq d;\text{ and
}A_{\infty}=H,$

where the Hörmander bracket condition $A_{\infty}=H$ is defined as in (1.4).

The first result is the following time inhomogeneous Ruelle-Perron-Frobenius
type theorem.

######  Theorem 2.1.

Let $V\in C_{b}^{1,0}(H\times\mathbb{T})$. Then there is a unique family of
eigen-triple $(\Gamma^{V},\lambda^{V},F^{V})$ for the Feynman-Kac evolution
operators $P_{0,t,h}^{V}$, such that

$(\Gamma^{V},\lambda^{V},F^{V})\in C(\mathbb{T},\mathcal{P}(H))\times
C(\mathbb{T},\mathbb{R})\times C(H\times\mathbb{T}),$

with $F^{V}$ strictly positive everywhere. Moreover, the following properties
hold:

  1. (1)

Normalization property:
$\left\langle\Gamma^{V}(h),F^{V}(\cdot,h)\right\rangle=1$ for all
$h\in\mathbb{T}$.
Eigenmeasure/eigenfunction property,

$\displaystyle
P_{0,t,h}^{V*}\Gamma^{V}(h)=\lambda_{t,h,V}\Gamma^{V}(\beta_{t}h),\qquad\text{(eigenmeasure
property)}$ $\displaystyle
P_{0,t,h}^{V}F^{V}(\cdot,\beta_{t}h)(w)=\lambda_{t,h,V}F^{V}(\cdot,h),\quad\text{(eigenfunction
property)}$

with positive eigenvalue

(2.10)
$\displaystyle\lambda_{t,h,V}=\exp\left(\int_{0}^{t}\lambda^{V}(\beta_{r}h)dr\right),\,\text{
and }\lambda^{V}(h)=\left\langle V(\cdot,h),\Gamma^{V}(h)\right\rangle,$

for all $h\in\mathbb{T}$ and $t\geq 0$.

  2. (2)

Stability property: for any $\eta\in(0,\eta_{0})$ and $\phi\in
C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$, and $M,R>0$, denoting

$\widetilde{\phi}(w,h)=\phi(w,h)-\left\langle\phi(\cdot,h),\Gamma^{V}(h)\right\rangle
F^{V}(w,h),\quad(w,h)\in H\times\mathbb{T},$

then we have for every $h\in\mathbb{T}$ and $R>0$,

    1. (a)

Pullback stability:

$\displaystyle\lim_{t\to\infty}\left|\lambda_{t,\beta_{-t}h,V}^{-1}P_{0,t,\beta_{-t}h}^{V}\widetilde{\phi}(\cdot,h)\right|_{C_{b}(B_{R}^{H})}=0,$
$\displaystyle\lim_{t\to\infty}\left|\lambda_{t,\beta_{-t}h,V}^{-1}P_{0,t,\beta_{-t}h}^{V}\widetilde{\phi}(\cdot,h)\right|_{\Gamma^{V}(\beta_{-t}h)}=0,$
$\displaystyle\lim_{t\to\infty}\sup_{\mu\in\mathcal{P}_{\eta^{\prime},M}(H)}\left|\left\langle\lambda_{t,\beta_{-t}h,V}^{-1}P_{0,t,\beta_{-t}h}^{V}\widetilde{\phi}(\cdot,h),\mu\right\rangle\right|=0,\,\forall\eta^{\prime}\in(\eta,\eta_{0});$

    2. (b)

Forward stability:

$\displaystyle\lim_{t\to\infty}\left|\lambda_{t,h,V}^{-1}P_{0,t,h}^{V}\widetilde{\phi}(\cdot,\beta_{t}h)\right|_{C_{b}(B_{R}^{H})}=0,$
$\displaystyle\lim_{t\to\infty}\left|\lambda_{t,h,V}^{-1}P_{0,t,h}^{V}\widetilde{\phi}(\cdot,\beta_{t}h)\right|_{\Gamma^{V}(h)}=0,$
(2.11)
$\displaystyle\lim_{t\to\infty}\sup_{\mu\in\mathcal{P}_{\eta^{\prime},M}(H)}\left|\left\langle\lambda_{t,h,V}^{-1}P_{0,t,h}^{V}\widetilde{\phi}(\cdot,\beta_{t}h),\mu\right\rangle\right|=0,\,\forall\eta^{\prime}\in(\eta,\eta_{0}).$

######  Remark 2.1.

Here are some remarks about the theorem.

  1. (1)

In fact, we have

$(\Gamma^{V},F^{V})\in
C(\mathbb{T},\mathcal{P}_{\mathfrak{m}_{\eta}}(H))\times
C_{\mathfrak{m}_{q}}(H\times\mathbb{T}),$

for $\eta\in(0,\eta_{0})$ and some $q=q(V)\geq 1$, where $\eta_{0}$ is from
Proposition B.1. See Proposition 3.1 and Proposition 3.2.

  2. (2)

If the potential $V=0$ then $(\Gamma^{V},\lambda^{V},F^{V})=(\Gamma,0,1)$,
where $\Gamma$ is the unique periodic invariant measure of the Markov
operators $P_{0,t,h}$ of system (2.1). In this vein, Theorem 2.1 extends the
unique ergodicity obtained in [28] to general Feynman-Kac evolution operators.

The second theorem is a Donsker-Varadhan type large deviation principle for
occupation measures of the time inhomogeneous solution processes. Denote the
occupation measure by

(2.12) $\displaystyle\zeta_{t,h}$
$\displaystyle=\frac{1}{t}\int_{0}^{t}\delta_{(\Phi_{0,r,h},\beta_{r}h)}dr,$

which is a family of random probability measures on $H\times\mathbb{T}$.

######  Theorem 2.2.

For any $\eta\in(0,\eta_{0})$ and $M>0$, the family of random probability
measures $\zeta_{t,h}$ satisfies the LDP for any $h\in\mathbb{T}$, uniformly
with respect to initial measures $\sigma\in\mathcal{P}_{\eta,M}(H)$, i.e.,
there is a non-trivial good rate function
$I:\mathcal{P}(H\times\mathbb{T})\to[0,\infty]$ that does not depend on
$\eta,M$ such that for any $h\in\mathbb{T}$, we have

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\log\sup_{\sigma\in\mathcal{P}_{\eta,M}(H)}\mathbf{P}_{\sigma}\\{\zeta_{t,h}\in
F\\}\leq-\inf_{\mu\in F}I(\mu),$
$\displaystyle\liminf_{t\to\infty}\frac{1}{t}\log\sup_{\sigma\in\mathcal{P}_{\eta,M}(H)}\mathbf{P}_{\sigma}\\{\zeta_{t,h}\in
G\\}\geq-\inf_{\mu\in G}I(\mu),$

for any closed set $F$ and open set $G$ in $\mathcal{P}(H\times\mathbb{T})$.

######  Remark 2.2.

Here are some remarks about the theorem.

  1. (1)

The rate function $I$ is the Legendre transformation of the pressure function
$Q$,

(2.15) $\displaystyle
I(\mu)=\left\\{\begin{array}[]{cc}\displaystyle\sup_{V\in
C_{b}(H\times\mathbb{T})}\left(\left\langle
V,\mu\right\rangle-Q(V)\right),&\mu\in\mathcal{P}(H\times\mathbb{T}),\\\
\qquad+\infty,&\text{otherwise.}\end{array}\right.$

Here the pressure function $Q:C_{b}(H\times\mathbb{T})\to\mathbb{R}$ is given
by the limit

(2.16) $\displaystyle
Q(V)=\lim_{t\to\infty}\frac{1}{t}\sup_{\sigma\in\mathcal{P}_{\eta,M}(H)}\log\mathbf{E}_{\sigma}\exp\left\langle
tV,\zeta_{t,h}\right\rangle,\text{ for }V\in C_{b}(H\times\mathbb{T}),$

where the limit $Q(V)$ does not depend on initial data
$\sigma\in\mathcal{P}_{\eta,M}(H)$ and $h\in\mathbb{T}$. The function $Q$ is
$1$-Lipschitz and convex. In particular,

$Q(V)=\int_{\mathbb{T}}\lambda^{V}(h)m(dh),\text{ for }V\in
C_{b}^{1,0}(H\times\mathbb{T}),$

where $\lambda^{V}\in C(\mathbb{T},\mathbb{R})$ is the eigenvalue obtained in
Theorem 2.1. See Theorem 4.1.

  2. (2)

For any Borel set $E\subset\mathbb{R}$ and $\phi\in C_{b}(H)$, define the set

$K(\phi_{\Gamma},E):=\left\\{\mu\in\mathcal{P}(H\times\mathbb{T}):\left\langle\phi_{\Gamma},\mu\right\rangle\in
E\right\\},$

where $\Gamma\in C(\mathbb{T},\mathcal{P}_{\mathfrak{m}_{\eta}}(H))$ is the
unique periodic invariant measure of the Navier-Stokes system (2.1) obtained
in [28] and
$\phi_{\Gamma}(w,h)=\phi(w)-\left\langle\phi,\Gamma(h)\right\rangle$. Note
that

$\displaystyle\zeta_{t,0}\in K(\phi_{\Gamma},E)\text{ is equivalent to
}\left\langle\phi_{\Gamma},\zeta_{t,0}\right\rangle=\frac{1}{t}\int_{0}^{t}\left(\phi(\Phi_{0,r})-\int_{H}\phi(w)\Gamma(\beta_{r}0)\right)dr\in
E.$

Thus by Theorem 2.2 one obtains a version of the LDP involving the unique
periodic invariant measure for solution processes of (2.1):

$\displaystyle\limsup_{t\to\infty}\frac{1}{t}\log\mathbf{P}_{\sigma}\left\\{\frac{1}{t}\int_{0}^{t}\left(\phi(\Phi_{0,r})-\int_{H}\phi(w)\Gamma(\beta_{r}0)\right)dr\in
E\right\\}\leq-L^{+}(\phi_{\Gamma},E),$
$\displaystyle\liminf_{t\to\infty}\frac{1}{t}\log\mathbf{P}_{\sigma}\left\\{\frac{1}{t}\int_{0}^{t}\left(\phi(\Phi_{0,r})-\int_{H}\phi(w)\Gamma(\beta_{r}0)\right)dr\in
E\right\\}\geq-L^{-}(\phi_{\Gamma},E),$

for any $\sigma\in\mathcal{P}_{\eta,M}(H)$, where

$\displaystyle L^{-}(\phi_{\Gamma},E)=\inf_{\mu\in
K(\phi_{\Gamma},\mathring{E})}I(\mu),\,\text{ and
}L^{+}(\phi_{\Gamma},E)=\inf_{\mu\in K(\phi_{\Gamma},\overline{E})}I(\mu).$

The central limit theorem proved in [28] implies that
$\sqrt{t}\left\langle\phi_{\Gamma},\zeta_{t,0}\right\rangle$ is asymptotically
Gaussian, which describes small deviations. Here the LDP gives quantitative
information on the probabilities of large deviations.

##  3\. A Ruelle-Perron-Frobenius type theorem

The aim of this section is to prove the Ruelle-Perron-Frobenius type Theorem
2.1 on the asymptotic behavior of the Feynman-Kac evolution operators
$P_{0,t,h}^{V}$ in terms of the corresponding eigen-triple. We first show the
existence of eigen-triple of the discrete time operators
$P_{0,k,h}^{V},k\in\mathbb{N}$ in the first subsection 3.1. Then in subsection
3.2 we prove the forward and pullback stability of the eigen-triple for
$P_{0,k,h}^{V}$. Subsection 3.3 is devoted to the proof of uniquenesses and
certain regularity properties of the eigen-triple. The proof of Theorem 2.1
will be given in subsection 3.4.

###  3.1. Existence of eigen-triple

In this subsection we establish the existence of eigen-triple for the discrete
time operators $P_{0,k,h}^{V},k\in\mathbb{N}$. In Proposition 3.1 we prove the
existence of eigenmeasure and eigenvalue by combining the idea of pullback and
a Schauder-Tychonoff’s fixed point theorem. Then in Proposition 3.2 we
construct the corresponding eigenfunction by utilizing the eigenmeasure and
eigenvalue, as well as a Krylov–Bogolyubov averaging procedure.

For $\eta,A>0$, let $\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$ be
the closure of $C(\mathbb{T},\mathcal{P}_{\eta,A}(H))$ with respect to the
topology of point-wise convergence. Using the idea of pull-back we define the
following operator family

(3.1) $\displaystyle
S_{t}^{V}(\gamma)(h):=\frac{P_{0,t,\beta_{-t}h}^{V*}\gamma(\beta_{-t}h)}{\big{(}P_{0,t,\beta_{-t}h}^{V*}\gamma(\beta_{-t}h)\big{)}(H)},\quad(t,h)\in\mathbb{R}_{+}\times\mathbb{T},\gamma\in\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}.$

It is straightforward to verify that this family of operators has the
semigroup property, whose fixed point gives the desired eigenmeasure.

######  Proposition 3.1 (Existence of Eigenmeasure and eigenvalue).

Assume $V\in C_{b}^{1,0}(H\times\mathbb{T})$ and $\eta\in(0,\eta_{0})$. Then
there is a constant $A>0$ such that the operator $S_{1}^{V}$ maps
$\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$ into itself, in which
it has a fixed point $\Gamma^{V}$. Denote
$\lambda_{k,h,V}=P_{0,k,h}^{V*}\Gamma^{V}(h)(H)$. Then $\Gamma^{V}$ has the
following properties:

  1. (1)

Eigenmeasure property:

$P_{0,k,h}^{V*}\Gamma^{V}(h)=\lambda_{k,h,V}\Gamma^{V}(\beta_{k}h),\,(k,h)\in\mathbb{N}\times\mathbb{T}.$

Cocycle property of eigenvalues $\lambda_{k,h,V}$:

(3.2)
$\displaystyle\lambda_{m+\ell,h,V}=\lambda_{m,\beta_{\ell}h,V}\lambda_{\ell,h,V},\text{
and }\lambda_{m,h,V}=\prod_{k=0}^{m-1}\lambda_{1,\beta_{k}h,V}.$

  2. (2)

Support property: for each $h\in\mathbb{T}$, the support of $\Gamma^{V}(h)$ is
$H$. In addition, for any $\rho,r>0$, there is a number $p=p(\rho,r,V)>0$ such
that

(3.3) $\displaystyle\inf_{(w,h)\in
B_{\rho}^{H_{2}}\times\mathbb{T}}\Gamma^{V}(h)(B_{r}^{H,H_{2}}(w))\geq p.$

Moreover, there is a large $L_{0}$ such that for any $L\geq L_{0}$, there
holds

(3.4) $\displaystyle\inf_{(w,h)\in
B_{\rho}^{H_{2}}\times\mathbb{T}}\Gamma^{V}(h)(B_{r}^{H,B_{L}^{H_{2}}}(w))\geq
p/2.$

  3. (3)

Concentration property: for each $h\in\mathbb{T}$, the measure $\Gamma^{V}(h)$
is concentrated on $H_{2}$, i.e. $\Gamma^{V}(h)(H_{2})=1$. There is a constant
$C$ such that for any $R>0$,

(3.5) $\displaystyle\sup_{h\in\mathbb{T}}\Gamma^{V}(h)(H\setminus
B_{R}^{H_{2}})\leq\frac{C}{R^{2}}.$

###### Proof.

The proof is divided into the following four steps.

Step 1: Show that $S_{1}^{V}$ maps $C(\mathbb{T},\mathcal{P}_{\eta,A}(H))$
into itself for appropriately chosen $A$. Let $\gamma\in
C(\mathbb{T},\mathcal{P}_{\eta,A}(H))$ and recall from the notation that
$m_{V},M_{V}$ the infimum and supremum of $V$. We first observe that

(3.6)
$\displaystyle\begin{split}0<e^{m_{V}}\leq&\big{(}P_{0,1,\beta_{-1}h}^{V*}\gamma(\beta_{-1}h)\big{)}(H)\leq
e^{M_{V}},\forall h\in\mathbb{T}.\end{split}$

By estimate (B.1) and Jensen’s inequality,

$\displaystyle\int_{H}e^{\eta\|w\|^{2}}\left(S_{1}^{V}(\gamma)(h)\right)(dw)$
$\displaystyle=\frac{\int_{H}\mathbf{E}\left[\exp\left(\int_{0}^{1}V\big{(}\mathcal{H}_{r}(w,\beta_{-1}h)\big{)}dr+\eta\|\Phi_{0,1,\beta_{-1}h}(w)\|^{2}\right)\right]\gamma(\beta_{-1}h)(dw)}{\big{(}P_{0,1,\beta_{-1}h}^{V*}\gamma(\beta_{-1}h)\big{)}(H)}$
$\displaystyle\leq
e^{M_{V}-m_{V}}\int_{H}\mathbf{E}e^{\eta\|\Phi_{0,1,\beta_{-1}h}(w)\|^{2}}\gamma(\beta_{-1}h)(dw)$
$\displaystyle\leq Ce^{M_{V}-m_{V}}\int_{H}e^{\eta
e^{-\nu}\|w\|^{2}}\gamma(\beta_{-1}h)(dw)$ $\displaystyle\leq
Ce^{M_{V}-m_{V}}\left(\int_{H}e^{\eta\|w\|^{2}}\gamma(\beta_{-1}h)(dw)\right)^{e^{-\nu}}\leq
CA^{e^{-\nu}}e^{(M_{V}-m_{V})},$

therefore if we choose $A\geq\exp\left(\frac{\ln
C+M_{V}-m_{V}}{1-e^{-\nu}}\right)$ then
$S_{1}^{V}(\gamma)(h)\in\mathcal{P}_{\eta,A}(H)$ for every $h\in\mathbb{T}$.

It remains to show the continuity, that is $S_{1}^{V}(\gamma)(h_{n})\to
S_{1}^{V}(\gamma)(h)$ weakly if $h_{n}\to h$. Let $\phi$ be any bounded
Lipschitz function on $H$. Since $V\in C_{b}^{1,0}(H\times\mathbb{T})$, and
$|\beta_{r}h_{n}-\beta_{r}h|=|h_{n}-h|$, one has

(3.7)
$\displaystyle\begin{split}&\left|P_{0,1,h_{n}}^{V}\phi(w)-P_{0,1,h}^{V}\phi(w)\right|\\\
&\begin{aligned}
\leq&\mathbf{E}\left|e^{\int_{0}^{1}V(\mathcal{H}_{r}(w,h_{n}))dr}\left(\phi(\Phi_{0,1,h_{n}}(w))-\phi(\Phi_{0,1,h}(w))\right)\right|\\\
&\qquad+\mathbf{E}\left|\left(e^{\int_{0}^{1}V(\mathcal{H}_{r}(w,h_{n}))dr}-e^{\int_{0}^{1}V(\mathcal{H}_{r}(w,h))dr}\right)\phi(\Phi_{0,1,h}(w))\right|\end{aligned}\\\
&\leq
C(V,\phi)\mathbf{E}\left(\|\Phi_{0,1,h_{n}}(w)-\Phi_{0,1,h}(w)\|+\int_{0}^{1}\left|V(\mathcal{H}_{r}(w,h_{n}))-V(\mathcal{H}_{r}(w,h))\right|dr\right)\\\
&\leq
C(V,\phi)\left(\mathbf{E}\|\Phi_{0,r,h_{n}}(w)-\Phi_{0,r,h}(w)\|+\kappa(|h_{n}-h|)\right)\leq
C(V,\phi)e^{\eta\|w\|^{2}}(|h_{n}-h|+\kappa(|h_{n}-h|)),\end{split}$

for some modulus of continuity $\kappa$, where we used estimate (B.4) and
Lipschitz continuity of $f$ in the last step. Hence

(3.8)
$\displaystyle\begin{split}\left|\Big{\langle}\gamma(h_{n}),P_{0,1,h_{n}}^{V}\phi-
P_{0,1,h}^{V}\phi\Big{\rangle}\right|&\leq
C\int_{H}e^{\eta\|w\|^{2}}\gamma(h_{n})(dw)(|h_{n}-h|+\kappa(|h_{n}-h|))\\\
&\leq C(|h_{n}-h|+\kappa(|h_{n}-h|)).\end{split}$

Since $P_{0,1,h}^{V}$ is Feller, $P_{0,1,h}^{V}\phi\in C_{b}(H)$. By
continuity of $\gamma$, we have

(3.9)
$\displaystyle\left|\Big{\langle}\gamma(h_{n})-\gamma(h),P_{0,1,h}^{V}\phi\Big{\rangle}\right|\to
0,\quad n\to\infty.$

Therefore by (3.8) and (3.9), we have

$\left|\Big{\langle}\gamma(h_{n}),P_{0,1,h_{n}}^{V}\phi\Big{\rangle}-\Big{\langle}\gamma(h),P_{0,1,h}^{V}\phi\Big{\rangle}\right|\to
0,\quad n\to\infty,$

which means $P_{0,1,h_{n}}^{V*}\gamma(h_{n})$ converges weakly to
$P_{0,1,h}^{V}\gamma(h)$ since $\phi$ was chosen arbitrarily. Hence

$h\to P_{0,1,h}^{V}\gamma(h)$

is continuous. As a result, $S_{1}^{V}(\gamma)(h)$ is continuous in
$h\in\mathbb{T}$. So $S_{1}^{V}$ maps $C(\mathbb{T},\mathcal{P}_{\eta,A}(H))$
into itself as desired.

Step 2: Prove that $S_{1}^{V}$ maps
$\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$ into itself. We claim
that $S_{1}^{V}$ is a continuous map from
$\mathcal{P}_{\eta,A}(H)^{\mathbb{T}}$ to itself with respect to the topology
of point-wise convergence. Then it will follow that $S_{1}^{V}$ maps
$\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$ into itself. Foy any
$h\in\mathbb{T}$, let $\Pi_{h}$ be the canonical projection of
$\mathcal{P}_{\eta,A}(H)^{\mathbb{T}}$ to the $h$ component, which is
$\mathcal{P}_{\eta,A}(H)$.

Note that $P_{0,t,\beta_{-1}h}^{V*}$ is a continuous map from
$\mathcal{P}_{\eta,A}(H)$ to itself, and

$F_{1}:\quad\gamma\to
P_{0,1,\beta_{-1}h}^{V*}\gamma(\beta_{-1}h)=P_{0,1,\beta_{-1}h_{0}}^{V*}\Pi_{\beta_{-1}h}\gamma$

is the composition of $P_{0,1,\beta_{-1}h}^{V*}$ with the projection
$\Pi_{\beta_{-1}h}$. Hence it is a continuous map from
$\mathcal{P}_{\eta,A}(H)^{\mathbb{T}}$ to $\mathcal{P}_{\eta,A}(H)$.
Similarly, we see that

$F_{2}:\quad\gamma\to\frac{1}{P_{0,1,\beta_{-1}h}^{V*}\gamma(\beta_{-1}h)(H)}$

is a continuous map from $\mathcal{P}_{\eta,A}(H)^{\mathbb{T}}$ to
$\mathbb{R}$ since $P_{0,1,\beta_{-1}h}^{V*}\gamma(\beta_{-1}h)(H)$ is bounded
away from zero by (3.6). Observe that the product map

$(a,\mu)\to a\mu$

is continuous, hence $\Pi_{h}S_{1}^{V}$ as the composition of the product map
with $(F_{2},F_{1})$ is continuous. Since $h$ is arbitrary, we have that
$S_{1}^{V}$ from $\mathcal{P}_{\eta,A}(H)^{\mathbb{T}}$ to itself is
continuous.

Step 3: Existence of a fixed point. We show that the image of $S_{1}^{V}$ is
precompact and conclude the existence of a fixed point by Schauder-Tychonoff’s
fixed point theorem. In view of the well-known Tychonoff’s compactness
theorem, to show precompactness, we only need to show that
$\left\\{(S_{1}^{V}\gamma)(h):\gamma\in\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}\right\\}$
is tight for every $h\in\mathbb{T}$. From estimate (B.6), one has

$\displaystyle\int_{H}\|w\|_{2}^{2}\left(S_{1}^{V}\gamma(h)\right)(dw)$
$\displaystyle=\frac{1}{P_{0,1,\beta_{-1}h}^{V*}\gamma(\beta_{-1}h)(H)}\int_{H}\mathbf{E}\left[\exp\left(\int_{0}^{1}V(\mathcal{H}_{r}(w,\beta_{-1}h))dr\right)\|\Phi_{0,1,\beta_{-1}h}(w)\|_{2}^{2}\right]\gamma(\beta_{-1}h)(dw)$
$\displaystyle\leq
e^{M_{V}-m_{V}}\int_{H}Ce^{\eta\|w\|^{2}}\gamma(\beta_{-1}h)(dw)\leq C.$

Hence

(3.10) $\displaystyle S_{1}^{V}\gamma(h)\\{\|w\|_{2}>R\\}\leq
R^{-2}\int_{H}\|w\|_{2}^{2}\left(S_{1}^{V}\gamma(h)\right)(dw)\leq R^{-2}C.$

Since $\\{\|w\|_{2}\leq R\\}$ is compactly embedded in $H$, we know that
$\left\\{(S_{1}^{V}\gamma)(h):\gamma\in\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}\right\\}$
is tight.

Now observe that $\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$ is a
nonempty closed convex subset of the locally convex Hausdorff topological
vector space of the maps from $\mathbb{T}$ to the space of finite signed
measures on $H$, and $S_{1}^{V}$ is a continuous mapping of
$\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$ into a compact subset.
Hence the Schauder-Tychonoff’s theorem (See Appendix of [5] or Theorem 10.1 in
[33]) implies the existence of a fixed point $\Gamma^{V}$ of $S_{1}^{V}$ in
$\overline{C_{p}(\mathbb{T},\mathcal{P}_{\eta,A}(H))}$.

Step 4: Prove that the fixed point has the desired properties in items
(1)-(3). The eigenmeasure property and cocycle property directly follow from
the definition of $S_{k}^{V}$. Indeed, Since $S_{k}^{V}$ has the semigroup
property, $\Gamma^{V}$ is also a fixed point of $S_{m}^{V}$, $m\in\mathbb{N}$,
i.e.,

$\displaystyle
P_{0,m,\beta_{-m}h}^{V*}\Gamma^{V}(\beta_{-m}h)=P_{0,m,\beta_{-m}h}^{V*}\Gamma^{V}(\beta_{-m}h)(H)\Gamma^{V}(h),\quad\forall
h\in\mathbb{T}.$

Replacing $h$ by $\beta_{m}h$, one obtains

$\displaystyle
P_{0,m,h}^{V*}\Gamma^{V}(h)=P_{0,m,h}^{V*}\Gamma^{V}(h)(H)\Gamma^{V}(\beta_{m}h),\quad\forall
h\in\mathbb{T}.$

Combining this with the cocycle property of the inhomogeneous Feynman-Kac
evolution operators, we have

$\displaystyle P_{0,m,h}^{V*}\Gamma^{V}(h)$
$\displaystyle=P_{0,m-1,\beta_{1}h}^{V*}P_{0,1,h}^{V*}\Gamma^{V}(h)=P_{0,m-2,\beta_{2}h}^{V*}P_{0,1,\beta_{1}h}^{V*}P_{0,1,h}^{V*}\Gamma^{V}(h)(H)\Gamma^{V}(\beta_{1}h)$
$\displaystyle=P_{0,m-3,\beta_{3}h}^{V*}P_{0,1,\beta_{2}h}^{V*}\Gamma^{V}(\beta_{2}h)P_{0,1,\beta_{1}h}^{V*}\Gamma^{V}(\beta_{1}h)(H)P_{0,1,h}^{V*}\Gamma^{V}(h)(H)$
$\displaystyle\cdots$
$\displaystyle=\prod_{k=0}^{m-1}P_{0,1,\beta_{k}h}^{V*}\Gamma^{V}(\beta_{k}h)(H)\Gamma^{V}(\beta_{m}h).$

The desired cocycle property then follows since $\Gamma^{V}(\beta_{m}h)(H)=1$.

The concentration property follows from (3.10). To show (3.3), we use the
uniform irreducibility from Proposition C.3. From the concentration inequality
(3.5), we can choose an $R$ large such that

$\inf_{h\in\mathbb{T}}\Gamma^{V}(h)(B_{R}^{H})\geq 1/2.$

Choose $m$ large such that $m>2\ln R$. Then it follows from (3.6) and
Proposition C.3 that

$\displaystyle\Gamma^{V}(h)(B_{r}^{H,H_{2}}(w))$
$\displaystyle=\int_{H}\mathbb{1}_{B_{r}^{H,H_{2}}(w)}(u)(S_{m}^{V}\Gamma^{V})(h)(dw)$
$\displaystyle=\int_{H}\mathbb{1}_{B_{r}^{H,H_{2}}(w)}(u)\frac{P_{0,m,\beta_{-m}h}^{V*}\Gamma^{V}(\beta_{-m}h)(du)}{\big{(}P_{0,m,\beta_{-m}h}^{V*}\Gamma^{V}(\beta_{-m}h)\big{)}(H)}$
$\displaystyle\geq
e^{-mM_{V}}\int_{H}P_{0,m,\beta_{-m}h}^{V}(u,B_{r}^{H,H_{2}}(w))\Gamma^{V}(\beta_{-m}h)(du)$
$\displaystyle\geq
e^{-mM_{V}}\int_{B_{R}^{H}}P_{0,m,\beta_{-m}h}^{V}(u,B_{r}^{H,H_{2}}(w))\Gamma^{V}(\beta_{-m}h)(du)\geq\frac{p}{2}e^{-mM_{V}},$

where $p=p(r,\rho)$ is from Proposition C.3. Hence (3.3) is proved. The proof
of (3.4) is similar by combining (C.18). In particular we have shown that
$\Gamma^{V}(h)$ has a positive measure for any nondegenerate ball in $H$, so
its support is $H$. The proof is complete. ∎

We now prove the existence of an eigenfunction corresponding to the eigenvalue
and eigenmeasure from the previous proposition. The scheme of the proof by
using a Krylov–Bogolyubov averaging procedure is a time inhomogeneous version
of the arguments developed in [23].

######  Proposition 3.2 (Existence of Eigenfunction).

Assume $A_{\infty}=H$ and $V\in C_{b}^{1,0}(H\times\mathbb{T})$. Then the
family of functions

$\displaystyle
F_{k}(w,h):=\frac{1}{k}\sum_{\ell=0}^{k-1}\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)}{\lambda_{\ell,h,V}},\quad(k,h)\in\mathbb{N}\times\mathbb{T}$

is uniformly equicontinuous on $(B_{R}^{H_{2}},\|\cdot\|)$ for any $R>0$. So
there is a subsequence converges to a function $F^{V}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$ for $q=q(V)$ as in
Proposition C.1. Moreover, $F^{V}$ satisfies the following

  1. (1)

Strict positivity: for every $(w,h)\in H\times\mathbb{T}$,

(3.11) $\displaystyle F^{V}(w,h)>0.$

  2. (2)

Eigenfunction property:

(3.12) $\displaystyle
P_{0,m,h}^{V}F^{V}(\cdot,\beta_{m}h)(w)=\lambda_{m,h,V}F^{V}(w,h),\quad(w,h)\in
H\times\mathbb{T},m\in\mathbb{N}.$

  3. (3)

Normalization property:

(3.13) $\displaystyle\langle F^{V}(\cdot,h),\Gamma^{V}(h)\rangle=1,\quad
h\in\mathbb{T}.$

  4. (4)

Equicontinuity: for any $R>0$ the restriction of the family
$\left\\{F^{V}(\cdot,h)\right\\}_{h\in\mathbb{T}}$ to
$(B_{R}^{H_{2}},\|\cdot\|)$ is equicontinuous.

###### Proof.

The proof is divided into the following three steps.

Step 1: Existence of $F^{V}\in
L_{\mathfrak{m}_{q}}^{\infty}(H_{2}\times\mathbb{T})$ satisfying items (3) and
(4). By uniform Feller property from Proposition C.4, we have that

$\displaystyle\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)}{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}=\frac{\quad\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)}{\lambda_{\ell,h,V}}\quad}{\quad\frac{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}{\lambda_{\ell,h,V}}\quad}$

is uniformly equicontinuous on $(B_{R}^{H_{2}},\|\cdot\|)$ for any $R>0$. If
one can show the existence of a constant $C=C(R_{0})$ such that

(3.14)
$\displaystyle\sup_{\ell,h}\frac{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}{\lambda_{\ell,h,V}}\leq
C,$

then $\\{F_{k}\\}$ will be uniformly equicontinuous.

Let
$g(w)=\displaystyle\inf_{\ell,h}\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)}{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}},w\in
H_{2}$. Since the infimum of a uniform equicontinuous and bounded family is
still continuous, $g\in C(B_{R}^{H_{2}},\|\cdot\|)$ for any $R>0$. Note that
$g$ is non-negative and each
$\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}}{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}$
is on the unit sphere in $C(B_{R_{0}}^{H_{2}},\|\cdot\|)$, hence there must be
a point $w_{0}\in B_{R_{0}}^{H_{2}}$ and a constant $a>0$ such that
$g(w_{0})>2a$. Then for each $R\geq R_{0}$, by the fact that $g\in
C(B_{R}^{H_{2}},\|\cdot\|)$, there is $r>0$ such that $g(w)>a$ for any $w\in
B_{r}^{H,B_{R}^{H_{2}}}(w_{0})$. By choosing $R$ large, the support property
(3.4) implies the existence of a constant $c>0$ such that

$\inf_{h}\Gamma^{V}(h)(B_{r}^{H,B_{R}^{H_{2}}}(w_{0}))\geq c.$

Since $\Gamma^{V}(h)$ is concentrated on $H_{2}$ for every $h\in\mathbb{T}$,
one has

$\displaystyle\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}^{-1}\lambda_{\ell,h,V}$
$\displaystyle=\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}^{-1}\int_{H}P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)\Gamma^{V}(h)(dw)$
$\displaystyle\geq\int_{H_{2}}g(w)\Gamma^{V}(h)(dw)\geq\int_{B_{r}^{H,B_{R}^{H_{2}}}(w_{0})}g(w)\Gamma^{V}(h)(dw)$
$\displaystyle\geq ca.$

This verifies the uniform bound (3.14). Hence
$\left\\{F_{k}(\cdot,h)\right\\}$ is uniformly equicontinuous on
$(B_{R}^{H_{2}},\|\cdot\|)$ for any $R>0$. It is also uniformly bounded on any
ball $(B_{R}^{H_{2}},\|\cdot\|)$ by the growth condition from Proposition C.1.

By the Arzelà–Ascoli theorem, there is a subsequence
$\\{F_{k_{j}}(\cdot,h)\\}$ and a function
$F^{V}(\cdot,h):H_{2}\to\mathbb{R}_{+}$ that is continuous when restricted on
$(B_{R}^{H_{2}},\|\cdot\|)$, such that

(3.15) $\displaystyle\lim_{j\to\infty}\sup_{w\in
B_{R}^{H_{2}}}|F_{k_{j}}(w,h)-F^{V}(w,h)|=0,\quad\forall R>0.$

The growth condition from Proposition C.1 implies that

$\displaystyle\frac{\qquad\frac{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{L_{\mathfrak{m}_{q}}^{\infty}}}{\lambda_{\ell,h,V}}\qquad}{\frac{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}{\lambda_{\ell,h,V}}}\leq\frac{\|P_{0,\ell,h}^{V}\mathfrak{m}_{q}\|_{L_{\mathfrak{m}_{q}}^{\infty}}}{\|P_{0,\ell,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}\leq
M_{q}.$

Together with boundedness (3.14), this shows that

(3.16)
$\displaystyle\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)}{\lambda_{\ell,h,V}}\leq
C_{R_{0}}M_{q}\mathfrak{m}_{q}(w),\quad w\in H,$

where the right-hand side is $\Gamma^{V}(h)$ integrable. Hence Lebesgue’s
dominated convergence theorem and eigenmeasure property from Proposition 3.1
yield

$\displaystyle\langle
F^{V}(\cdot,h),\Gamma^{V}(h)\rangle=\lim_{j\to\infty}\langle
F_{k_{j}}(\cdot,h),\Gamma^{V}(h)\rangle=1,\quad\forall h\in\mathbb{T},$

which verifies the normalization property (3.13). It also follows from (3.15)
and (3.16) that $F^{V}\in
L_{\mathfrak{m}_{q}}^{\infty}(H_{2}\times\mathbb{T})$. Since for each $R>0$,
the family $\\{F_{k}(\cdot,h)\\}_{h\in\mathbb{T}}$ is uniformly equicontinuous
on the ball $(B_{R}^{H_{2}},\|\cdot\|)$, one deduces item (4) about
equicontinuity of $\\{F^{V}(\cdot,h)\\}_{h\in\mathbb{T}}$.

Step 2: Proof of $F^{V}\in L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$
and item (2) on the eigenfunction property. We first show that the
eigenfunction property is true for $w\in H_{2}$. By cocycle property (3.2),

$\displaystyle(\mathcal{P}_{1}F_{k_{j}})(w,h)$
$\displaystyle=\frac{1}{k_{j}}\sum_{\ell=0}^{k_{j}-1}P_{0,1,h}^{V}\left(\frac{P_{0,\ell,\beta_{1}h}^{V}\mathbb{1}_{H}(\cdot)}{\lambda_{\ell,\beta_{1}h,V}}\right)(w)$
$\displaystyle=\frac{\lambda_{1,h,V}}{k_{j}}\sum_{\ell=0}^{k_{j}-1}\frac{P_{0,\ell+1,h}^{V}\mathbb{1}_{H}(w)}{\lambda_{1,h,V}\lambda_{\ell,\beta_{1}h,V}}$
$\displaystyle=\frac{\lambda_{1,\beta_{1}h,V}}{k_{j}}\sum_{\ell=0}^{k_{j}-1}\frac{P_{0,\ell,h}^{V}\mathbb{1}_{H}(w)}{\lambda_{\ell,h,V}}-\frac{\lambda_{1,\beta_{1}h,V}}{k_{j}}\left(1+\frac{P_{0,k_{j},h}^{V}\mathbb{1}_{H}(w)}{\lambda_{k_{j},h,V}}\right)$

It follows from the construction of $F^{V}(w,h)$ and the boundedness from
(3.16) that the right-hand side converges to
$\lambda_{1,\beta_{1}h,V}F^{V}(w,h)$ as $j\to\infty$. The left-hand side
converges to

$\mathcal{P}_{1}F^{V}(w,h)=P_{0,1,h}^{V}F^{V}(\cdot,\beta_{1}h)(w)$

by dominated convergence theorem and the fact that $w_{0,1,h}\in H_{2}$. This
proves the eigenfunction property for $m=1$ and $(w,h)\in
H_{2}\times\mathbb{T}$. The general case for $m\geq 2$ and $(w,h)\in
H_{2}\times\mathbb{T}$ follows from cocycle property (3.2). By the
regularizing property of the Navier-Stokes solution process, for any $(w,h)\in
H\times\mathbb{T}$, the function
$\frac{P_{0,1,h}^{V}F^{V}(\cdot,\beta_{1}h)(w)}{\lambda_{1,h,V}}$ is well
defined and satisfies (similar to the proof of (3.16))

$\displaystyle\frac{P_{0,1,h}^{V}F^{V}(\cdot,\beta_{1}h)(w)}{\lambda_{1,h,V}}\leq
C(F^{V},R_{0})M_{q}\mathfrak{m}_{q}(w),\quad w\in H.$

Define
$F^{V}(w,h)=\frac{P_{0,1,h}^{V}F^{V}(\cdot,\beta_{1}h)(w)}{\lambda_{1,h,V}}$
for those $(w,h)\in H\times\mathbb{T}\backslash H_{2}\times\mathbb{T}$. Then
$F^{V}\in L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$ is a nonnegative
function satisfying the eigenfunction property for any $(w,h)\in
H\times\mathbb{T}$. Since $\Gamma^{V}(h)$ is concentrated on $H_{2}$, the
definition on $H\setminus H_{2}$ does not affect the normalization identity
(3.13).

Step 3: Proof of item (1) on the strict positivity. Let $R\geq 1$, and
$(w_{0},h_{0})\in B_{R}^{H}\times\mathbb{T}$. Since $\left\langle
F^{V}(\cdot,h),\Gamma^{V}(h)\right\rangle=1$, and $\Gamma^{V}(h)$ is
concentrated on $H_{2}$, for every $h\in\mathbb{T}$ there is a point $w_{h}\in
H_{2}$ and a number $c=c(w_{h})>0$ such that $F^{V}(w_{h},h)\geq c$. To apply
the irreducibility from Proposition C.3, we choose an $R>0$ such that
$w_{0}\in B_{R}^{H}$, and integer $\ell>2\ln R$. Then we choose
$L>L_{0}(\ell,R)$ as in (C.18) large such that $w_{\beta_{\ell}h_{0}}\in
B_{L}^{H_{2}}$. Since $F^{V}(\cdot,\beta_{\ell}h_{0})$ is continuous on
$(B_{L}^{H_{2}},\|\cdot\|)$, there is a radius $r>0$ such that

$F^{V}(w,\beta_{\ell}h_{0})\geq c/2\text{ for any }w\in
B_{r}^{H,B_{L}^{H_{2}}}(w_{\beta_{\ell}h_{0}}).$

By the eigenfunciton property (3.12) and boundedness of $V$, we have,

$\displaystyle
F^{V}(w_{0},h_{0})=\frac{P^{V}_{0,\ell,h_{0}}F^{V}(w_{0},\beta_{\ell}h_{0})}{\lambda_{\ell,h_{0},V}}$
$\displaystyle\geq e^{-\ell
M_{V}}\int_{B_{r}^{H,B_{L}^{H_{2}}}(w_{\beta_{\ell}h_{0}})}F^{V}(w,\beta_{\ell}h_{0})P^{V}_{0,\ell,h_{0}}(w_{0},du)$
$\displaystyle\geq ce^{-\ell
M_{V}}P^{V}_{0,\ell,h_{0}}(w_{0},B_{r}^{H,B_{L}^{H_{2}}}(w_{\beta_{\ell}h_{0}}))>0$

by (C.18).

The proof of Proposition 3.2 is complete. ∎

###  3.2. Forward and pullback stability for discrete time

This subsection is devoted to the proof of the forward stability in the
following Theorem 3.3 for the eigen-triple of the discrete time Feynman-Kac
evolution operators $P_{0,k,h}^{V}$. The pullback stability is stated in
Theorem 3.6 at the end of the section where the proof is omitted since it is
similar to the proof of forward stability by combining the
eigenfunction/eigenmeasure property in the pullback sense.

######  Theorem 3.3 (Forward stability the eigen-triple).

Let $\eta\in(0,\eta_{0})$ and $V\in C_{b}^{1,0}(H\times\mathbb{T})$. For any
$\phi\in C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$, $M,R>0,h\in\mathbb{T}$,
and $\eta^{\prime}\in(\eta,\eta_{0})$ we have

(3.17)
$\displaystyle\lim_{k\to\infty}\left|\lambda_{k,h,V}^{-1}P_{0,k,h}^{V}\phi(\cdot,\beta_{k}h)-\left\langle\phi(\cdot,\beta_{k}h),\Gamma^{V}(\beta_{k}h)\right\rangle
F^{V}(\cdot,h)\right|_{C_{b}(B_{R}^{H})}=0,$ (3.18)
$\displaystyle\lim_{k\to\infty}\left|\lambda_{k,h,V}^{-1}P_{0,k,h}^{V}\phi(\cdot,\beta_{k}h)-\left\langle\phi(\cdot,\beta_{k}h),\Gamma^{V}(\beta_{k}h)\right\rangle
F^{V}(\cdot,h)\right|_{\Gamma^{V}(h)}=0,\,$ (3.19)
$\displaystyle\lim_{k\to\infty}\sup_{\mu\in\mathcal{P}_{\eta^{\prime},M}(H)}\left|\left\langle\lambda_{k,h,V}^{-1}P_{0,k,h}^{V*}\mu,\varphi(\cdot,\beta_{k}h)\right\rangle-\langle
F^{V}(\cdot,h),\mu\rangle\left\langle\Gamma^{V}(\beta_{k}h),\varphi(\cdot,\beta_{k}h)\right\rangle\right|=0,$

This theorem is proved in subsection 3.2.2, after we establish the
$L^{1}$-forward stability for differentiable observable functions in
subsection 3.2.1 through a version of Ruelle’s lower bound technique. Before
proceeding, let’s introduce some notations and a lemma.

Fix $h\in\mathbb{T}$ for which the dependence on $h$ will be dropped unless
stated throughout the proof. Set

(3.20) $\displaystyle
g(w,h)=\phi(w,h)-\left\langle\phi(\cdot,h),\Gamma^{V}(h)\right\rangle
F^{V}(w,h)$

and

(3.21) $\displaystyle
g_{k}(w,h)=\frac{(\mathcal{P}_{k}^{V}g)(w,h)}{\lambda_{k,h,V}}.$

It follows from the eigenfunction property (3.12) that for every
$h\in\mathbb{T}$,

(3.22) $\displaystyle
g_{k}(w,h)=\frac{P_{0,k,h}^{V}\phi(\cdot,\beta_{k}h)(w)}{\lambda_{k,h,V}}-\left\langle\phi(\cdot,\beta_{k}h),\Gamma^{V}(\beta_{k}h)\right\rangle
F^{V}(w,h).$

and

(3.23) $\displaystyle\left\langle
g_{k}(\cdot,h),\Gamma^{V}(h)\right\rangle=0,\text{ and }\left\langle
g_{k}^{+}(\cdot,h),\Gamma^{V}(h)\right\rangle=\left\langle
g_{k}^{-}(\cdot,h),\Gamma^{V}(h)\right\rangle=\frac{1}{2}|g_{k}(\cdot,h)|_{\Gamma^{V}(h)}.$

by combining the eigenmeasure property from Proposition 3.1. The following
observation is useful.

######  Lemma 3.4.

Let $V\in C_{b}^{1,0}(H\times\mathbb{T})$, $\phi\in
C_{b}^{1}(H\times\mathbb{T})$ and $q$ be the number from Proposition 3.2. Then
the family $g_{k}(\cdot,h)$ satisfies

$\displaystyle\\{g_{k}(\cdot,h)\\}_{(k,h)\in\mathbb{N}\times\mathbb{T}}\subset
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T}),$

and is uniformly equicontinuous on $C_{b}(B_{R}^{H_{2}},\|\cdot\|)$ for any
$R>0$.

###### Proof.

In view of (3.16) and the fact that $\phi\in C_{b}^{1}(H\times\mathbb{T})$, we
have

$\displaystyle\frac{P_{0,k,h}^{V}\phi(\cdot,\beta_{k}h)(w)}{\lambda_{k,h,V}}\leq\|\phi\|_{\infty}\frac{P_{0,k,h}^{V}\mathbb{1}_{H}(w)}{\lambda_{k,h,V}}\leq
C\mathfrak{m}_{q}(w),$

where the constant $C$ does not depend on $h,k$. In addition,
$\left\langle\phi(\cdot,\beta_{k}h),\Gamma^{V}(\beta_{k}h)\right\rangle$ is
bounded uniformly for $(k,h)\in\mathbb{N}\times\mathbb{T}$. It then follows
from Proposition 3.2 that the family

$\left\\{\left\langle\phi(\cdot,\beta_{k}h),\Gamma^{V}(\beta_{k}h)\right\rangle
F^{V}(\cdot,h)\right\\}_{(k,h)\in\mathbb{N}\times\mathbb{T}}$

is uniformly equicontinuous on $B_{R}^{H_{2}}$ and

$\left|\left\langle\phi(\cdot,\beta_{k}h),\Gamma^{V}(\beta_{k}h)\right\rangle
F^{V}(w,h)\right|\leq C\mathfrak{m}_{q}(w).$

As a consequence (note that the following (3.24) is still true if $\phi\in
C_{b}(H\times\mathbb{T})$),

(3.24) $\displaystyle\sup_{(k,h)\in\mathbb{N}\times\mathbb{T}}|g_{k}(w,h)|\leq
C\mathfrak{m}_{q}(w).$

Furthermore, the uniform bound (3.14) and the uniform Feller property in
Proposition C.4 imply that the family

$\left\\{\frac{P_{0,k,h}^{V}\phi(\cdot,\beta_{k}h)}{\lambda_{k,h,V}}\right\\}_{(k,h)\in\mathbb{N}\times\mathbb{T}}$

is also uniformly equicontinuous on $(B_{R}^{H_{2}},\|\cdot\|)$. Therefore so
is $\\{g_{k}(\cdot,h)\\}_{(k,h)\in\mathbb{N}\times\mathbb{T}}$.

∎

####  3.2.1. Forward stability in $L^{1}$ norm

This subsection is devoted to prove the following proposition on forward
stability in $L^{1}$ norm for differentiable observable functions.

######  Proposition 3.5.

For $V\in C_{b}^{1,0}(H\times\mathbb{T})$, $\phi\in
C_{b}^{1}(H\times\mathbb{T})$, and $h\in\mathbb{T}$, we have

(3.25)
$\displaystyle\lim_{k\rightarrow\infty}\left|\lambda_{k,h,V}^{-1}P_{0,k,h}^{V}\phi\left(\cdot,\beta_{k}h\right)-\left\langle\phi\left(\cdot,\beta_{k}h\right),\Gamma^{V}\left(\beta_{k}h\right)\right\rangle
F^{V}(\cdot,h)\right|_{\Gamma^{V}(h)}=0.$

###### Proof.

We prove the convergence by contradiction. The proof is divided into four
steps.

Step 1: Lower bounds from contradiction assumption. Assume (3.25) does not
hold. Then there is a constant $a>0$ and a subsequence $k_{j}\to\infty$ such
that

(3.26) $\displaystyle|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}\geq a.$

Due to the time inhomogeneity, here we do not have non-expansiveness of the
sequence $g_{k}(\cdot,h)$ under $L^{1}(\Gamma^{V}(h))$ norm. Instead, the
cocycle property (3.2) implies

(3.27) $\displaystyle
g_{k}(w,h)=\frac{P^{V}_{0,l,h}g_{k-l}(\cdot,\beta_{l}h)(w)}{\lambda_{l,h,V}},\,\text{
for all }0\leq l\leq k,$

which together with the eigenmeasure property from Proposition 3.1 yields

(3.28)
$\displaystyle|g_{k}(\cdot,h)|_{\Gamma^{V}(h)}\leq|g_{k-l}(\cdot,\beta_{l}h)|_{\Gamma^{V}(\beta_{l}h)},\,\text{
for all }0\leq l\leq k,h\in\mathbb{T}.$

This “local” monotonicity along the line $x+y=k$ and (3.26) imply that

$\displaystyle|g_{m}(\cdot,\beta_{n}h)|_{\Gamma^{V}(\beta_{n}h)}\geq
a,\,\text{ for all }(m,n)\in K,$

where $K=\bigcup_{j\geq 1}K_{j}$ with

$\displaystyle K_{j}=\left\\{(m,n)\in\mathbb{Z}^{2}:m,n\geq 0\text{ and
}m+n=k_{j}\right\\}.$

Combining with (3.23), we have

(3.29)
$\displaystyle|g_{m}^{\pm}(\cdot,\beta_{n}h)|_{\Gamma^{V}(\beta_{n}h)}\geq
a/2,\,\text{ for all }(m,n)\in K.$

Since $\Gamma^{V}(h)$ is concentrated on $H_{2}$ for every $h$, we have from
(3.5) and Lemma 3.4 that

$\displaystyle|g_{m}^{\pm}(\cdot,\beta_{n}h)|_{\Gamma^{V}(\beta_{n}h)}$
$\displaystyle=\int_{H_{2}}g_{m}^{\pm}(w,\beta_{n}h)\Gamma^{V}(\beta_{n}h)(dw)$
$\displaystyle\leq\int_{B_{L_{1}}^{H_{2}}}g_{m}^{\pm}(w,\beta_{n}h)\Gamma^{V}(\beta_{n}h)(dw)+C\int_{H_{2}\setminus
B_{L_{1}}^{H_{2}}}\mathfrak{m}_{q}(w)\Gamma^{V}(\beta_{n}h)(dw)$
$\displaystyle\leq\int_{B_{L_{1}}^{H_{2}}}g_{m}^{\pm}(w,\beta_{n}h)\Gamma^{V}(\beta_{n}h)(dw)+a/4,$

for some large $L_{1}$ independent of $m,n,h$. This and (3.29) imply the
existence of $w_{m,n}^{\pm}\in B_{L_{1}}^{H_{2}}$ such that

$\displaystyle g_{m}^{\pm}(w_{m,n}^{\pm},\beta_{n}h)\geq a/4$

for all $(m,n)\in K$. Let $r_{0}$ be the radius from (C.19) and
$\ell_{0}=\lceil 2\ln r_{0}\rceil$, and $L_{0}=L_{0}(\ell_{0},r_{0})$ as in
Proposition C.3. Since the family
$\left\\{g_{m}^{\pm}(,\beta_{n}h)\right\\}_{(m,n)\in K}$ is uniformly
equicontinuous on $(B_{L_{0}}^{H_{2}},\|\cdot\|)$ by Lemma 3.4, there is a
radius $r=r(L_{0})$ independent of $m,n,h$ such that

(3.30) $\displaystyle g_{m}^{\pm}(w,\beta_{n}h)\geq a/8,\text{ for all }w\in
B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{m,n}^{\pm}).$

By (C.18) we have $p_{0}=p_{0}(L_{1},r)$ independent of $m,n$ such that

$\displaystyle
P^{V}_{0,\ell_{0},h}(w_{0},B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{m,n}^{\pm}))\geq\inf_{(w_{0},h,w)\in
B_{r_{0}}^{H_{2}}\times\mathbb{T}\times
B_{L_{1}}^{H_{2}}}P^{V}_{0,\ell_{0},h}(w_{0},B_{r}^{H,B_{L_{0}}^{H_{2}}}(w))\geq
p_{0}/2.$

for all $m,n\in K$. Hence for any $\ell\geq 0$, by the Chapman-Kolmogorov
relation, we have

$\displaystyle
P^{V}_{0,\ell+\ell_{0},h}(w,B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{m,n}^{\pm}))$
$\displaystyle=\int_{H}P^{V}_{\ell,\ell+\ell_{0},h}(u,B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{m,n}^{\pm}))P^{V}_{0,\ell,h}(w,du)$
$\displaystyle\geq\int_{B_{r_{0}}^{H_{2}}}P^{V}_{0,\ell_{0},\beta_{\ell}h}(u,B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{m,n}^{\pm}))P^{V}_{0,\ell,h}(w,du\geq\frac{1}{2}p_{0}P^{V}_{0,\ell,h}(w,B_{r_{0}}^{H_{2}})$

for all $w\in H$. As a result, for any $R\geq 1$, by taking $\ell=\lceil 2\ln
R\rceil+\ell_{0}$, one obtains by (C.19) that

(3.31) $\displaystyle
P^{V}_{0,\ell,h}(w,B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{m,n}^{\pm}))\geq\frac{1}{2}p_{0}P^{V}_{0,\ell-\ell_{0},h}(w,B_{r_{0}}^{H_{2}})\geq
p_{0}/4,\,\forall(w,h)\in B_{R}^{H}\times\mathbb{T},(m,n)\in K.$

Step 2: A local Harnack type inequality from the lower bounds. Let $R\geq 1$
and $\ell=\lceil 2\ln R\rceil+\ell_{0}$. Then for each $j\geq 1$, we can write
$k_{j}={l}_{j}\ell+r_{j}$ where $l_{j},r_{j}$ are nonnegative integers such
that $l_{j}\to\infty$ as $j\to\infty$ and $r_{j}<\ell$. We now establish an
estimate which roughly states that the action of the Feynman-Kac evolution
operators on $g_{m}^{\pm}(\cdot,\beta_{n}h)$ satisfies a local Harnack type
inequality as long as $(m,n)\in K$. See [25, 23] in the time homogeneous case.

Fix any $j$ such that $l_{j}\geq 2$. Note that for any $0\leq p<l_{j}$, we
have $(k_{j}-(p+1)\ell,(p+1)\ell)\in K_{j}$. Let
$w_{0}^{\pm}=w^{\pm}_{k_{j}-(p+1)\ell,(p+1)\ell}$ be the corresponding points
as in (3.30). Assume without loss of generality that $V\geq 0$ and recall that
$\displaystyle M_{V}=\sup_{w\in H}V(w)$. Then by boundedness of $V$, and
estimates (3.30), (3.31),

$\displaystyle\begin{split}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}&=\lambda_{\ell,\beta_{p\ell}h,V}^{-1}\int_{H}g_{k_{j}-(p+1)\ell}^{\pm}(u,\beta_{(p+1)\ell}h)P^{V}_{0,\ell,\beta_{p\ell}h}(w,du)\\\
&\geq e^{-\ell
M_{V}}\int_{B_{r}^{H,B_{L_{0}}^{H_{2}}}(w_{0}^{\pm})}g_{k_{j}-(p+1)\ell}^{\pm}(u,\beta_{(p+1)\ell}h)P^{V}_{0,\ell,\beta_{p\ell}h}(w,du)\\\
&\geq Ce^{-\ell M_{V}},\end{split}$

for all $w\in B_{R}^{H}$, where $C=p_{0}a/32$ is independent of
$p,\ell,R,k_{j}$. On the other hand, from (3.14),(3.24) and the growth
condition from Proposition C.1 we deduce

(3.32)
$\displaystyle\begin{split}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}&\leq
C\lambda_{\ell,\beta_{p\ell}h,V}^{-1}P^{V}_{0,\ell,\beta_{p\ell}h}\mathfrak{m}_{q}(w)\\\
&\leq
C\lambda_{\ell,\beta_{p\ell}h,V}^{-1}\|P^{V}_{0,\ell,\beta_{p\ell}h}\mathbb{1}_{H}\|_{R_{0}}\mathfrak{m}_{q}(w)\\\
&\leq C\mathfrak{m}_{q}(R),\end{split}$

for all $w\in B_{R}^{H}$, where the constant $C\geq 1$ is independent of
$p,\ell,R,k_{j}$. As a result, we obtain a local Harnack type inequality,

$\displaystyle\sup_{w\in
B^{H}_{R}}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}\leq
A(R)\inf_{w\in
B^{H}_{R}}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}},$

where since $\ell=\lceil 2\ln R\rceil+\ell_{0}$,

(3.33) $\displaystyle A(R)=C\mathfrak{m}_{q}(R)e^{\ell M_{V}}\leq
CR^{2M_{V}}\mathfrak{m}_{q}(R),\quad C\text{ a constant independent of
}p,\ell,R,k_{j}.$

It then follows by the eigenmeasure property from Proposition 3.1 that

$\displaystyle|g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)|_{\Gamma^{V}(\beta_{(p+1)\ell}h)}$
$\displaystyle=\int_{H}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}\Gamma^{V}(\beta_{p\ell}h)(dw)$
$\displaystyle\leq A(R)\inf_{w\in
B^{H}_{R}}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}+I_{0},$

where

(3.34) $\displaystyle\begin{split}I_{0}&=\int_{H\setminus
B^{H}_{R}}\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}\Gamma^{V}(\beta_{p\ell}h)(dw)\\\
&\leq C\sup_{h\in\mathbb{T}}\int_{H\setminus
B^{H}_{R}}\mathfrak{m}_{q}(w)\Gamma^{V}(h)(dw):=\varepsilon_{0}(R)\end{split}$

by following the same estimates as in (3.32). Hence we arrive at

(3.35)
$\displaystyle\frac{P^{V}_{0,\ell,\beta_{p\ell}h}g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)(w)}{\lambda_{\ell,\beta_{p\ell}h,V}}-\frac{|g_{k_{j}-(p+1)\ell}^{\pm}(\cdot,\beta_{(p+1)\ell}h)|_{\Gamma^{V}(\beta_{(p+1)\ell}h)}}{A(R)}+\frac{\varepsilon_{0}(R)}{A(R)}\geq
0,$

for all $R\geq 1$, $\ell=\lceil 2\ln R\rceil+\ell_{0}$, $w\in B_{R}^{H},j\geq
1,0\leq p<l_{j}$, where $k_{j}=l_{j}\ell+r_{j}$.

Step 3: An iteration scheme based on the local Harnack type inequality. Now we
prove that $|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}$ can be made as arbitrarily
small by taking $j$ sufficiently large through an iteration scheme by
utilizing the estimate (3.35). It follows from (3.27), (3.23) and (3.34) that

(3.36)
$\displaystyle\begin{split}&|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}=\left|\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}\right|_{\Gamma^{V}(h)}\\\
&\leq\int_{B^{H}_{R}}\left|\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}(\cdot,\beta_{\ell}h)(w)}{\lambda_{\ell,h,V}}\right|\Gamma^{V}(h)(dw)+\varepsilon_{0}(R)\\\
&\begin{aligned}
=\int_{B^{H}_{R}}\Big{(}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{+}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}&-\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{-}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}\\\
&-\frac{|g_{k_{j}-\ell}^{+}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}+\frac{|g_{k_{j}-\ell}^{-}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}\Big{)}\Gamma^{V}(h)(dw)+\varepsilon_{0}(R)\end{aligned}\\\
&\begin{aligned}
\leq\int_{B^{H}_{R}}\Big{(}\Big{|}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{+}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}&-\frac{|g_{k_{j}-\ell}^{+}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}\Big{|}\\\
&+\Big{|}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{-}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}-\frac{|g_{k_{j}-\ell}^{-}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}\Big{|}\Big{)}\Gamma^{V}(h)(dw)+\varepsilon_{0}(R).\end{aligned}\end{split}$

Since for $c\geq 0$, inequality $a\geq-c$ implies $|a|\leq a+2c$, we have by
(3.35) with $p=0$ that

$\displaystyle\int_{B^{H}_{R}}\Big{|}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{\pm}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}-\frac{|g_{k_{j}-\ell}^{\pm}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}\Big{|}\Gamma^{V}(h)(dw)$
$\displaystyle\leq\int_{B^{H}_{R}}\Big{(}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{\pm}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}-\frac{|g_{k_{j}-\ell}^{\pm}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}\Big{)}\Gamma^{V}(h)(dw)+2A(R)^{-1}\varepsilon_{0}(R)$
$\displaystyle=\int_{B^{H}_{R}}\Big{(}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{\pm}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}\Big{)}\Gamma^{V}(h)(dw)-\frac{|g_{k_{j}-\ell}^{\pm}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}}{A(R)}\Gamma^{V}(h)(B_{R}^{H})+2A(R)^{-1}\varepsilon_{0}(R).$

Combining this estimate with (3.36), and the eigenfunction property from
Proposition 3.1 and (3.23), we deduce

(3.37)
$\displaystyle\begin{split}|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}\leq&\int_{H}\Big{(}\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{+}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}+\frac{P^{V}_{0,\ell,h}g_{k_{j}-\ell}^{-}(\cdot,\beta_{\ell}h)}{\lambda_{\ell,h,V}}\Big{)}\Gamma^{V}(h)(dw)\\\
&-\frac{\Gamma^{V}(h)(B_{R}^{H})}{A(R)}\Big{(}|g_{k_{j}-\ell}^{+}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}+|g_{k_{j}-\ell}^{-}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}\Big{)}+4A(R)^{-1}\varepsilon_{0}(R)+\varepsilon_{0}(R)\\\
&\leq
a_{0}(R)|g_{k_{j}-\ell}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}+\kappa(R)\end{split}$

where

(3.38) $\displaystyle a_{0}(R)=1-A(R)^{-1}\Gamma^{V}(h)(B^{H}_{R}),\text{ and
}\kappa(R)=(4A(R)^{-1}+1)\varepsilon_{0}(R).$

Apply the same estimates as in (3.36)-(3.37) to
$|g_{k_{j}-\ell}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}$ by using
(3.35) with $p=1$ we have

$\displaystyle|g_{k_{j}-\ell}(\cdot,\beta_{\ell}h)|_{\Gamma^{V}(\beta_{\ell}h)}\leq
a_{1}(R)|g_{k_{j}-2\ell}(\cdot,\beta_{2\ell}h)|_{\Gamma^{V}(\beta_{2\ell}h)}+\kappa(R),$

where

$a_{1}(R)=1-A(R)^{-1}\Gamma^{V}(\beta_{\ell}h)(B^{H}_{R}).$

By iteration through using (3.35) for $0\leq p<l_{j}$, one obtains

(3.39)
$\displaystyle|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}\leq\delta_{l_{j}-1}|g_{r_{j}}(\cdot,\beta_{l_{j}\ell}h)|_{\Gamma^{V}(\beta_{l_{j}\ell}h)}+\eta_{l_{j}}\kappa(R)\leq
C\delta_{l_{j}-1}+\eta_{l_{j}}\kappa(R),$

where

$\displaystyle\delta_{m}=\prod_{i=0}^{m}a_{i},\,\eta_{l_{j}}=1+\sum_{m=0}^{l_{j}-2}\delta_{m},\,\text{
and }a_{i}=a_{i}(R)=1-A(R)^{-1}\Gamma^{V}(\beta_{i\ell}h)(B^{H}_{R}),$

and $C$ does not depend on the parameters $k_{j},h,\ell,R,l_{j},r_{j}$. Note
that in the last inequality of (3.39), we used Lemma 3.4 and the integrability
of $\mathfrak{m}_{q}$ with respect to $\Gamma^{V}$ from Proposition 3.1.

Step 4: Deriving a contradiction. By the concentration property (3.5), we can
find $R_{0}>0$ such that for all $R\geq R_{0}$,

$\Gamma^{V}(h)(B^{H}_{R})\geq 1/2,\quad\forall h\in\mathbb{T}.$

Therefore

$\displaystyle 0<a_{i}\leq 1-\frac{1}{2}A(R)^{-1}:=\delta(R)<1.$

Combining this with (3.39), the formula of $\kappa(R)$ from (3.38), $A(R)$
from (3.33) and $\varepsilon_{0}(R)$ from (3.34), we know

(3.40) $\displaystyle\begin{split}|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}&\leq
C\delta(R)^{l_{j}-1}+\frac{\kappa(R)}{1-\delta(R)}=C\delta(R)^{l_{j}-1}+2(4+A(R))\varepsilon_{0}(R)\\\
&\leq
C\left(\delta(R)^{l_{j}-1}+R^{2M_{V}}\mathfrak{m}_{q}(R)\sup_{h\in\mathbb{T}}\int_{H\setminus
B_{R}^{H}}\mathfrak{m}_{q}(w)\Gamma^{V}(h)(dw)\right),\end{split}$

for any $R\geq R_{0}$. By Proposition 3.1 and Markov’s inequality, we find

(3.41) $\displaystyle\begin{split}\int_{H\setminus
B_{R}^{H}}\mathfrak{m}_{q}(w)\Gamma^{V}(h)(dw)&\leq\left(\int_{H}\mathfrak{m}_{q}^{2}(w)\Gamma^{V}(h)(dw)\right)^{1/2}\left(\Gamma^{V}(h)(H\setminus
B_{R}^{H})\right)^{1/2}\\\ &\leq
C\left(\Gamma^{V}(h)\left\\{\|w\|^{6(M_{V}+q)}\geq
R^{6(M_{V}+q)}\right\\}\right)^{1/2}\\\
&\leq\frac{C}{R^{3(M_{V}+q)}}.\end{split}$

Combining (3.40) and (3.41), for any given $\varepsilon>0$, one can first
choose $R$ large to make

$R^{2M_{V}}\mathfrak{m}_{q}(R)\sup_{h\in\mathbb{T}}\int_{H\setminus
B_{R}^{H}}\mathfrak{m}_{q}(w)\Gamma^{V}(h)(dw)<\frac{\varepsilon}{2C},$

then choose $j$ large such that

$\delta(R)^{l_{j}-1}<\frac{\varepsilon}{2C},$

which result in $|g_{k_{j}}(\cdot,h)|_{\Gamma^{V}(h)}<\varepsilon$. This
contradicts with (3.26). As a result, convergence (3.25) must be true. ∎

####  3.2.2. Proof of Theorem 3.3

In this subsection we complete the proof of the forward stability as in
Theorem 3.3, by combining the $L^{1}$-stability established in Proposition
3.5, the uniform equicontinuity and standard approximation arguments.

###### Proof of Theorem 3.3.

We first show that Proposition 3.5 implies (3.17) for $\phi\in
C_{b}^{1}(H\times\mathbb{T})$. Then we prove them for $\phi\in
C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$ through approximation arguments.
The convergence (3.19) is given at the end. The proof is divided into the
following four steps.

Step 1: Convergence (3.17) in $C_{b}(B_{R}^{H_{2}},\|\cdot\|)$ for $\phi\in
C_{b}^{1}(H\times\mathbb{T})$. This is proved by a contradiction argument.
Indeed, if there is $R>0$ such that convergence (3.17) in
$C_{b}(B_{R}^{H_{2}},\|\cdot\|)$ does not hold, then there are $a,r>0$ and a
subsequence $k_{j}$ and $\\{w_{k_{j}}\\}\subset B_{R}^{H_{2}}$ such that

$|g_{k_{j}}(w,h)|\geq a>0,\forall w\in
B_{r}^{H,B_{L}^{H_{2}}}(w_{k_{j}}),\quad k\in\mathbb{N}$

by uniform equicontinuity. Here $r>0$ is independent of $h,k$ and $L$ is any
large constant from (3.4). The support property (3.4) then implies that

$\displaystyle|g_{k_{j}}(w,h)|_{\Gamma^{V}(h)}\geq
a\Gamma^{V}(h)(B_{r}^{H,B_{L}^{H_{2}}}(w_{k_{j}}))\geq a\inf_{(w,h)\in
B_{R}^{H_{2}}\times\mathbb{T}}\Gamma^{V}(h)(B_{r}^{H,B_{L}^{H_{2}}}(w))>0$

for all $j\geq 1$, which contradicts with the convergence (3.18).

The proof of Step 1 is complete.

Step 2: Convergence of (3.17) for $\phi\in C_{b}^{1}(H\times\mathbb{T})$ in
$C_{b}(B_{R}^{H})$. First observe that by (3.27), one has for any $R>0$,

$\displaystyle|g_{k+1}(w,h)|$
$\displaystyle=\left|\int_{H}g_{k}(u,\beta_{1}h)\frac{P^{V}_{0,1,h}(w,du)}{\lambda_{0,1,h}}\right|$
$\displaystyle\leq\int_{B_{R}^{H_{2}}}\left|g_{k}(u,\beta_{1}h)\right|\frac{P^{V}_{0,1,h}(w,du)}{\lambda_{0,1,h}}+\int_{H\setminus
B_{R}^{H_{2}}}\left|g_{k}(u,\beta_{1}h)\right|\frac{P^{V}_{0,1,h}(w,du)}{\lambda_{0,1,h}}$
$\displaystyle:=I_{1}+I_{2}.$

By (3.16) and the uniform convergence $\displaystyle\sup_{u\in
B_{R}^{H_{2}}}|g_{k}(u,\beta_{1}h)|\to 0$ from Step 1, one has for any $r>0$,

$\displaystyle\sup_{w\in B_{r}^{H}}I_{1}$ $\displaystyle\leq\sup_{u\in
B_{R}^{H_{2}}}|g_{k}(u,\beta_{1}h)|\sup_{w\in
B_{r}^{H}}\frac{P^{V}_{0,1,h}\mathbb{1}_{H}(w)}{\lambda_{0,1,h}}$
$\displaystyle\leq C_{R_{0}}M_{q}\mathfrak{m}_{q}(r)\sup_{u\in
B_{R}^{H_{2}}}|g_{k}(u,\beta_{-k}h)|\to 0,\quad k\to\infty.$

Now it follows from (3.24), (B.6) and Markov’s inequality that

$\displaystyle I_{2}$ $\displaystyle\leq C\int_{H\setminus
B_{R}^{H_{2}}}\mathfrak{m}_{q}(u)\frac{P^{V}_{0,1,h}(w,du)}{\lambda_{0,1,h}}$
$\displaystyle\leq
CR^{-1}\left(\mathbf{E}_{w}\mathfrak{m}_{q}^{2}(\Phi_{0,1,\beta_{1}h})\right)^{1/2}\left(\mathbf{E}_{w}\|\Phi_{0,1,\beta_{1}h}\|_{2}^{2}\right)^{1/2}\leq
CR^{-1}e^{\eta\|w\|^{2}}.$

Hence

$\sup_{w\in B_{r}^{H}}I_{2}\to 0,\quad R\to\infty.$

As a consequence, we deduce for any $r>0$,

$\sup_{w\in B_{r}^{H}}|g_{k}(w,h)|\to 0,\quad k\to\infty.$

The proof of Step 2 is complete.

Step 3: Convergence (3.17) and (3.18) for $\phi\in
C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$. We only show the proof of these
convergences for $\phi\in C_{b}(H\times\mathbb{T})$. The proof for $\phi\in
C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$ through a cut-off procedure is
similar.

By Theorem A.8 in [14], the space $C_{b}^{1}(H)$ is dense in $C_{b}(H)$ when
restricting on any compact subsets of $H$. Therefore by viewing
$\\{\phi(\cdot,\beta_{k}h)\\}_{k\geq 0}$ as an element in
$\ell^{\infty}(\mathbb{N},C_{b}(H))$, there is a sequence
$\phi_{m}=\\{\phi_{m}(\cdot,\beta_{k}h)\\}_{k\geq
0}\in\ell^{\infty}(\mathbb{N},C_{b}(H))$ such that $\displaystyle\sup_{m\geq
1}\|\phi_{m}\|_{\ell^{\infty}(\mathbb{N},C_{b}(H))}<\infty$ and
$\phi_{m}(\cdot,\beta_{k}h)\to\phi(\cdot,\beta_{k}h)$ uniformly for $k$ and
uniformly on compact subsets of $H$ as $m\to\infty$. Denote by $g_{k,m}$ the
function obtained by replacing $\phi(\cdot,\beta_{k}h)$ with
$\phi_{m}(\cdot,\beta_{k}h)$ in (3.22). And denote by
$\displaystyle\|\varphi\|_{r}=\sup_{w\in B_{r}^{H}}|\varphi(w)|$. Then in view
of (3.22), we have

$\displaystyle\|g_{k}(\cdot,h)\|_{r}\leq\|g_{k,m}(\cdot,h)\|_{r}$
$\displaystyle+\sup_{h\in\mathbb{T}}\|F^{V}(\cdot,h)\|_{r}\left\langle|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|,\Gamma^{V}(\beta_{k}h)\right\rangle$
$\displaystyle+\frac{\|P_{0,k,h}^{V}(\phi-\phi_{m})(\cdot,\beta_{k}h)\|_{r}}{\lambda_{k,h,V}}.$

It follows from Proposition 3.2 that
$\displaystyle\sup_{h\in\mathbb{T}}\|F^{V}(\cdot,h)\|_{r}<\infty$. Also from
(3.5), we have for any $R\geq 1$,

$\displaystyle\left\langle|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|,\Gamma^{V}(\beta_{k}h)\right\rangle$
$\displaystyle=\left\langle|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|\mathbb{1}_{B_{R}^{H_{2}}},\Gamma^{V}(\beta_{k}h)\right\rangle+\left\langle|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|\mathbb{1}_{H\setminus
B_{R}^{H_{2}}},\Gamma^{V}(\beta_{k}h)\right\rangle$
$\displaystyle\leq|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|_{C_{b}(B_{R}^{H_{2}})}+C\left\langle\mathbb{1}_{H\setminus
B_{R}^{H_{2}}},\Gamma^{V}(\beta_{k}h)\right\rangle$
$\displaystyle\leq|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|_{C_{b}(B_{R}^{H_{2}})}+CR^{-2},$

where $C$ is independent of $k,m,R$. Therefore for any $\varepsilon>0$, by
first choosing $R$ large and then choose $m$ large, we can find $M_{1}>0$ such
that for all $m\geq M_{1}$ and $k\geq 0$,

(3.42)
$\displaystyle\left\langle|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|,\Gamma^{V}(\beta_{k}h)\right\rangle<\varepsilon/3.$

Let $L\geq 1$. By the uniform bounds (3.14), (3.16) and the growth condition
for the function $N_{2}$ in Proposition C.1, we have for $w\in B_{r}^{H}$,

$\displaystyle\frac{|P_{0,k,h}^{V}(\phi-\phi_{m})(\cdot,\beta_{k}h)(w)|}{\lambda_{k,h,V}}$
$\displaystyle\leq\frac{|P_{0,k,h}^{V}\mathbb{1}_{B_{L}^{H_{2}}}(\phi-\phi_{m})(\cdot,\beta_{k}h)(w)|}{\lambda_{k,h,V}}+\frac{|P_{0,k,h}^{V}\mathbb{1}_{H\setminus
B_{L}^{H_{2}}}(\phi-\phi_{m})(\cdot,\beta_{k}h)(w)|}{\lambda_{k,h,V}}$
$\displaystyle\leq|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|_{C_{b}(B_{L}^{H_{2}})}\frac{\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{r}}{\lambda_{k,h,V}}+C\frac{\left\|P_{0,k,h}^{V}\mathbb{1}_{H\setminus
B_{L}^{H_{2}}}\right\|_{r}}{\lambda_{k,h,V}}$ $\displaystyle\leq
C\left(|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|_{C_{b}(B_{L}^{H_{2}})}+L^{-2}\frac{\left\|P_{0,k,h}^{V}N_{2}\right\|_{r}}{\lambda_{k,h,V}}\right)$
$\displaystyle\leq
C\left(|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|_{C_{b}(B_{L}^{H_{2}})}+L^{-2}\frac{\left\|P_{0,k,h}^{V}\mathbb{1}_{H}\right\|_{R_{0}}}{\lambda_{k,h,V}}\right)$
$\displaystyle\leq
C(|\phi(\cdot,\beta_{k}h)-\phi_{m}(\cdot,\beta_{k}h)|_{C_{b}(B_{L}^{H_{2}})}+L^{-2}),$

where the constant $C$ does not depend on $L,k,m$. Hence by first choosing $L$
large and then choosing $m$ large, we can find $M_{2}>0$ such that for all
$m\geq M_{2}$ and $k\geq 0$

(3.43) $\displaystyle\sup_{w\in
B_{r}^{H}}\frac{\|P_{0,k,h}^{V}(\phi-\phi_{m})(\cdot,\beta_{k}h)\|_{r}}{\lambda_{k,h,V}}<\varepsilon/3.$

The result from Step 2 implies that for $m=\max\\{M_{1},M_{2}\\}$, one can
find $K>0$ such that for all $k\geq K$,

$\|g_{k,m}(\cdot,h)\|_{r}<\varepsilon/3.$

Combining this with (3.42) and (3.43) we obtain the desired convergence (3.17)
for $\phi\in C_{b}(H\times\mathbb{T})$. The convergence (3.18) follows from
(3.17) and the concentration property (3.5) and the fact that
$\Gamma^{V}(h)\in\mathcal{P}_{\eta,A}(H)$ from Proposition 3.1.

The proof of Step 3 is complete.

Step 4: Proof of convergence (3.19). Let $\phi\in
C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$ and
$\eta^{\prime}\in(\eta,\eta_{0})$. Then by the uniform bound (3.14) and growth
condition from Proposition C.1, we have

$\displaystyle\frac{P_{0,k,h}^{V}\phi(w,h)}{\lambda_{k,h,V}}$
$\displaystyle\leq
C\frac{P_{0,k,h}^{V}\mathfrak{m}_{\eta}(w)}{\lambda_{k,h,V}}$
$\displaystyle=C\frac{P_{0,k,h}^{V}\mathfrak{m}_{\eta}(w)}{\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}\frac{\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}{\lambda_{k,h,V}}\leq
C\mathfrak{m}_{\eta}(w).$

Since $F^{V}\in L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$, one has
$g_{k}\in L_{\mathfrak{m}_{\eta}}^{\infty}(H\times\mathbb{T})$, where $g_{k}$
is the function obtained from (3.21) with $\phi\in
C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$. Hence by Hölder’s inequality and
Markov inequality, one has

(3.44)
$\displaystyle\begin{split}\left\langle|g_{k}(\cdot,h)|,\mu\right\rangle&=\left\langle|g_{k}(\cdot,h)|\mathbb{1}_{B_{R}^{H}},\mu\right\rangle+\left\langle|g_{k}(\cdot,h)|\mathbb{1}_{H\setminus
B_{R}^{H}},\mu\right\rangle\\\
&\leq|g_{k}(\cdot,h)|_{C_{b}(B_{R}^{H})}+C\left\langle\mathfrak{m}_{\eta}\mathbb{1}_{H\setminus
B_{R}^{H}},\mu\right\rangle\\\
&\leq|g_{k}(\cdot,h)|_{C_{b}(B_{R}^{H})}+C\left\langle\mathfrak{m}_{\eta^{\prime}},\mu\right\rangle^{\eta/\eta^{\prime}}\left\langle\mathbb{1}_{H\setminus
B_{R}^{H}},\mu\right\rangle^{1-\eta/\eta^{\prime}}\\\
&\leq|g_{k}(\cdot,h)|_{C_{b}(B_{R}^{H})}+CR^{-1}\end{split}$

for all $\mu\in\mathcal{P}_{\eta^{\prime},M}(H)$, where $C$ is a constant
independent of $k,h,R$. Therefore by first taking $k\to\infty$ and then
$R\to\infty$, we obtain the desired convergence (3.19).

The proof of Theorem 3.3 is complete. ∎

Following the same idea, one can prove the following pullback stability. Note
that since we do not have uniform in $h\in\mathbb{T}$ convergence in Theorem
3.3, the following theorem is not a direct consequence of it. However, since
in the pullback case we have a nice global monotonicity structure, the proof
is actually simpler than that of Theorem 3.3.

######  Theorem 3.6 (Pullback stability of the eigen-triple).

Let $\eta\in(0,\eta_{0})$ and $V\in C_{b}^{1,0}(H\times\mathbb{T})$. For any
$\phi\in C_{\mathfrak{m}_{\eta}}(H\times\mathbb{T})$, $M,R>0,h\in\mathbb{T}$,
and $\eta^{\prime}\in(\eta,\eta_{0})$ we have

(3.45)
$\displaystyle\lim_{k\to\infty}\left|\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V}\phi(\cdot,h)-\left\langle\phi(\cdot,h),\Gamma^{V}(h)\right\rangle
F^{V}(\cdot,\beta_{-k}h)\right|_{C_{b}(B_{R}^{H})}=0,$ (3.46)
$\displaystyle\lim_{k\to\infty}\left|\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V}\phi(\cdot,h)-\left\langle\phi(\cdot,h),\Gamma^{V}(h)\right\rangle
F^{V}(\cdot,\beta_{-k}h)\right|_{\Gamma^{V}(\beta_{-k}h)}=0,$ (3.47)
$\displaystyle\lim_{k\to\infty}\sup_{\mu\in\mathcal{P}_{\eta^{\prime},M}(H)}\left|\left\langle\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V*}\mu,\phi(\cdot,h)\right\rangle-\langle
F^{V}(\cdot,\beta_{-k}h),\mu\rangle\left\langle\Gamma^{V}(h),\phi(\cdot,h)\right\rangle\right|=0.$

###### Proof.

Replacing $h$ by $\beta_{-k}h$ in (3.28), we have

$\displaystyle|g_{k}(\cdot,\beta_{-k}h)|_{\Gamma^{V}(\beta_{-k}h)}\leq|g_{k-1}(\cdot,\beta_{-(k-1)}h)|_{\Gamma^{V}(\beta_{-(k-1)}h)},\,\text{
for all }k,h\in\mathbb{T}.$

This means for each fixed $h$, the sequence
$|g_{k}(\cdot,\beta_{-k}h)|_{\Gamma^{V}(\beta_{-k}h)},\,k\in\mathbb{N}$ is
non-increasing. This global non-expansive structure implies that if the
convergence (3.46) does not hold, then there is a constant $a>0$ such that

$|g_{k}(\cdot,\beta_{-k}h)|_{\Gamma^{V}(\beta_{-k}h)}\geq a$

for all $k$. The rest of the proof for Theorem 3.6 is similar to that of
Theorem 3.3, hence we omit the details. ∎

The following corollary states that uniformly for initial measures in

$\overline{\Gamma}:=\left\\{\Gamma^{V}(h),h\in\mathbb{T}\right\\},$

the trajectory of $S_{k}^{V}$ is attracted by $\Gamma^{V}$, which is useful in
proving the continuity of eigenmeasure.

######  Corollary 3.7.

For each $V\in C_{b}^{1,0}(H\times\mathbb{T}),\phi\in C_{b}(H)$, we have the
following convergence

$\displaystyle\lim_{k\to\infty}\sup_{\mu\in\overline{\Gamma}}\left|\frac{\left\langle
P_{0,k,\beta_{-k}h}^{V*}\mu,\phi\right\rangle}{P_{0,k,\beta_{-k}h}^{V*}\mu(H)}-\left\langle\Gamma^{V}(h),\phi\right\rangle\right|=0,$

for every $h\in\mathbb{T}$.

###### Proof.

Let $\phi\in C_{b}(H)$ and $\eta\in(0,\eta_{0})$. Convergence (3.47) implies
that

(3.48) $\displaystyle
I_{1}:=\left|\left\langle\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V*}\mu,\phi\right\rangle-\langle
F^{V}(\cdot,\beta_{-k}h),\mu\rangle\left\langle\Gamma^{V}(h),\phi\right\rangle\right|\to
0$

uniformly for $\mu\in\mathcal{P}_{\eta,A}(H)$ as $k\to\infty$, where $A$ is
from Proposition 3.1. In particular, taking $\phi=\mathbb{1}_{H}$, then

(3.49) $\displaystyle
I_{2}:=\left|\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V*}\mu(H)-\left\langle
F^{V}(\cdot,\beta_{-k}h),\mu\right\rangle\right|\to 0$

uniformly for $\mu\in\mathcal{P}_{\eta,A}(H)$ as well. Combining (3.48) and
(3.49) with triangle inequality, we have

(3.50) $\displaystyle\varepsilon_{k,h,\mu}:=\left|\frac{\left\langle
P_{0,k,\beta_{-k}h}^{V*}\mu,\phi\right\rangle}{\lambda_{k,\beta_{-k}h,V}}-\frac{P_{0,k,\beta_{-k}h}^{V*}\mu(H)}{\lambda_{k,\beta_{-k}h,V}}\left\langle\Gamma^{V}(h),\phi\right\rangle\right|\leq
I_{1}+I_{2}\left|\left\langle\Gamma^{V}(h),\phi\right\rangle\right|\to 0$

uniformly for $\mu\in\mathcal{P}_{\eta,A}(H)$ as $k\to\infty$.

We now claim that
$\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V*}\mu(H)$ can be bounded
away from 0 for large $k$ through the convergence (3.49) and equicontinuity of
the eigenfunction family $F^{V}(\cdot,h),h\in\mathbb{T}$. Since
$\Gamma^{V}(h)$ is concentrated on $H_{2}$ for every $h$, by the normalizing
equality (3.13) and the fact $F^{V}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$ from Proposition 3.2, we
have for any $L>0$

$\displaystyle 1=\left\langle F^{V}(\cdot,h),\Gamma^{V}(h)\right\rangle$
$\displaystyle=\left\langle
F^{V}(\cdot,h)\mathbb{1}_{B_{L}^{H_{2}}},\Gamma^{V}(h)\right\rangle+\left\langle
F^{V}(\cdot,h)\mathbb{1}_{H_{2}\setminus
B_{L}^{H_{2}}},\Gamma^{V}(h)\right\rangle$ $\displaystyle\leq\left\langle
F^{V}(\cdot,h)\mathbb{1}_{B_{L}^{H_{2}}},\Gamma^{V}(h)\right\rangle+\left\langle\mathfrak{m}_{q}\mathbb{1}_{H_{2}\setminus
B_{L}^{H_{2}}},\Gamma^{V}(h)\right\rangle$ $\displaystyle\leq\left\langle
F^{V}(\cdot,h)\mathbb{1}_{B_{L}^{H_{2}}},\Gamma^{V}(h)\right\rangle+CL^{-1}.$

In the last inequality we used Hölder inequality, the concentration property
(3.5) and $\Gamma^{V}(h)$ integrability of $\mathfrak{m}_{q}^{2}$, where $C$
is independent of $h$. Taking $L\geq L_{0}$ sufficiently large where $L_{0}$
is from (3.4), we have

$\left\langle
F^{V}(\cdot,h)\mathbb{1}_{B_{L}^{H_{2}}},\Gamma^{V}(h)\right\rangle\geq
1/2,\,h\in\mathbb{T}.$

This implies the existence of $w_{h}\in B_{L}^{H_{2}}$ such that
$F^{V}(w_{h},h)\geq 1/2$. Proposition 3.2 implies that the family
$F^{V}(\cdot,h),h\in\mathbb{T}$ is uniformly equicontinuous on
$(B_{L}^{H_{2}},\|\cdot\|)$. Hence there is a radius $r=r(L)$ independent of
$h$ such that

$\displaystyle F^{V}(w,h)\geq 1/4,\quad\text{ for all }w\in
B_{r}^{H,B_{L}^{H_{2}}}(w_{h}),h\in\mathbb{T}.$

Combining this with (3.4), we have

(3.51) $\displaystyle\left\langle
F^{V}(\cdot,\beta_{-k}h),\mu\right\rangle\geq\frac{1}{4}\mu(B_{r}^{H,B_{L}^{H_{2}}}(w_{h}))\geq
p/8,$

for any $\mu\in\overline{\Gamma}$, $k\geq 0,h\in\mathbb{T}$, where $p=p(L)$ is
independent of $h,k,\mu$. In view of (3.49), we obtain the existence of a
large $K$ (may depend on $h$) such that

(3.52)
$\displaystyle\inf_{\mu\in\overline{\Gamma}}\lambda_{k,\beta_{-k}h,V}^{-1}P_{0,k,\beta_{-k}h}^{V*}\mu(H)\geq
p/16.$

for all $k\geq K$. Combining this with (3.50) and the fact
$\overline{\Gamma}\subset\mathcal{P}_{\eta,A}(H)$, we obtain the desired
convergence

$\displaystyle\sup_{\mu\in\overline{\Gamma}}\left|\frac{\left\langle
P_{0,k,\beta_{-k}h}^{V*}\mu,\phi\right\rangle}{P_{0,k,\beta_{-k}h}^{V*}\mu(H)}-\left\langle\Gamma^{V}(h),\phi\right\rangle\right|$
$\displaystyle=\sup_{\mu\in\overline{\Gamma}}\frac{\lambda_{k,\beta_{-k}h,V}}{P_{0,k,\beta_{-k}h}^{V*}\mu(H)}\varepsilon_{k,h,\mu}$
$\displaystyle\leq\frac{16}{p}\sup_{\mu\in\overline{\Gamma}}\varepsilon_{k,h,\mu}\to
0,\,k\to\infty.$

The proof is complete. ∎

###  3.3. Uniqueness and regularity of the eigen-triple

We now prove uniqueness of eigen-triple and corresponding regularity. The
uniqueness of eigenmeasure and eigenvalue follows from the pullback
convergence in Theorem 3.6, while the uniqueness of eigenfunction follows from
the forward convergence in Theorem 3.3. The continuous dependence of
eigenmeassure on time symbols $h\in\mathbb{T}$ follows from Corollary 3.7,
which implies continuity of eigenvalue $\lambda_{k,h,V}$. The regularity of
eigenfunction $F^{V}$ on $w\in H$ follows from the forward convergence (3.17)
and uniform Lipschitz property from Proposition C.4. Continuity of
eigenfunction on time symbols $h\in\mathbb{T}$ will be proved in subsection
3.4.

######  Proposition 3.8.

For $V\in C_{b}^{1,0}(H\times\mathbb{T})$, the eigen-triple
$(\Gamma^{V},\lambda_{k,h,V},F^{V})$ of the Feynman-Kac evolution operator
$P_{0,k,h}^{V}$ is unique. In addition,

$h\to\Gamma^{V}(h),\,\text{ and }h\to\lambda_{k,h,V},$

are continuous for every $k$. Moreover, the family
$\left\\{F^{V}(\cdot,h),h\in\mathbb{T}\right\\}$ is uniformly Lipschitz on
$B_{R}^{H}$ for any $R>0$ with a uniform bound $L=L(R,V)$ on the Lipschitz
constants independent of $h\in\mathbb{T}$.

###### Proof.

The proof is divided into three steps. We first show uniqueness at the first
step. Then we prove continuity of eigenmeasure and eigenvalue at the second
step. The continuity of eigenfunction is given at the last step.

Step 1: Uniqueness. Suppose there is another eigen-triple
$(\widetilde{\Gamma}^{V},\widetilde{\lambda}_{k,h,V},\widetilde{F}^{V})$. We
show the uniqueness by first proving $\Gamma^{V}=\widetilde{\Gamma}^{V}$.

Noting $\widetilde{\Gamma}^{V}$ must have the same support property as
$\Gamma^{V}$ in Proposition 3.1. This fact together with the convergence
(3.45) imply that

$\displaystyle\lim_{k\to\infty}|g_{k}(\cdot,\beta_{-k}h)|_{\widetilde{\Gamma}^{V}(\beta_{-k}h)}=0,$

where $g_{k}$ is given as in (3.21). Since $\left|\left\langle
g_{k}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle\right|\leq|g_{k}(\cdot,\beta_{-k}h)|_{\widetilde{\Gamma}^{V}(\beta_{-k}h)}$,
it follows that

(3.53) $\displaystyle\lim_{k\to\infty}\left\langle
g_{k}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle=0.$

Taking in particular $\phi=\mathbb{1}_{H}$ in $g_{k}$, using the facts
$\widetilde{\Gamma}^{V}(H)=\Gamma^{V}(H)=1$ for all $h\in\mathbb{T}$ and
$P_{0,k,\beta_{-k}h}^{V*}\widetilde{\Gamma}^{V}(\beta_{-k}h)=\widetilde{\lambda}_{k,\beta_{-k}h,V}\widetilde{\Gamma}^{V}(h)$,
we have

(3.54) $\displaystyle\left\langle
g_{k}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle=\frac{\widetilde{\lambda}_{k,\beta_{-k}h,V}}{\lambda_{k,\beta_{-k}h,V}}-\left\langle
F^{V}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle\to
0,\quad k\to\infty,$

by (3.53). Similarly by exchanging the role of eigen-triple we have

(3.55)
$\displaystyle\frac{\lambda_{k,\beta_{-k}h,V}}{\widetilde{\lambda}_{k,\beta_{-k}h,V}}-\left\langle\widetilde{F}^{V}(\cdot,\beta_{-k}h),\Gamma^{V}(\beta_{-k}h)\right\rangle\to
0,\quad k\to\infty,$

Observe that $F^{V}$ is strictly positive everywhere by Proposition 3.2. We
claim that

$\liminf_{k\to\infty}\left\langle
F^{V}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle>0.$

Indeed if this is not true, then by (3.54) we have

$\liminf_{k\to\infty}\frac{\widetilde{\lambda}_{k,\beta_{-k}h,V}}{\lambda_{k,\beta_{-k}h,V}}=0.$

This contradicts with the limit (3.55), since as an eigenfunction
$\widetilde{F}^{V}\in L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$
implies that
$\left\langle\widetilde{F}^{V}(\cdot,\beta_{-k}h),\Gamma^{V}(\beta_{-k}h)\right\rangle$
is uniformly bounded. Hence the claim is proved.

Combining the claim with (3.54), one has

$\displaystyle\Lambda_{k}:=\frac{\widetilde{\lambda}_{k,\beta_{-k}h,V}}{\lambda_{k,\beta_{-k}h,V}}\left\langle
F^{V}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle^{-1}\to
1,\quad k\to\infty,$

which together with (3.53) imply that for every $\phi\in C_{b}(H)$,

$\displaystyle\left|\left\langle\phi,\widetilde{\Gamma}^{V}(h)\right\rangle-\left\langle\phi,\Gamma^{V}(h)\right\rangle\right|$
$\displaystyle=\lim_{k\to\infty}\left|\Lambda_{k}\left\langle\phi,\widetilde{\Gamma}^{V}(h)\right\rangle-\left\langle\phi,\Gamma^{V}(h)\right\rangle\right|$
$\displaystyle=\lim_{k\to\infty}\left\langle
F^{V}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle^{-1}\left|\left\langle
g_{k}(\cdot,\beta_{-k}h),\widetilde{\Gamma}^{V}(\beta_{-k}h)\right\rangle\right|=0.$

Hence the eigenmeasure, as the fixed point of the semigroup $S_{k}^{V}$ in
Proposition 3.1, is unique. The corresponding eigenvalue $\lambda_{k,h,V}$ is
uniquely determined. Furthermore, the eigenfunction $F^{V}$ satisfying the
normalization condition $\left\langle F^{V}(h),\Gamma^{V}(h)\right\rangle=1$
is also uniquely determined by the forward convergence in Theorem 3.3 by
taking $\phi=\mathbb{1}_{H}$ in (3.17).

Step 2: Continuity of eigenmeasure and eigenvalue. Let $\phi$ be any bounded
Lipschitz function on $H$ and $h_{0}\in\mathbb{T}$. We would like to show

$\left\langle\Gamma^{V}(h_{n}),\phi\right\rangle\to\left\langle\Gamma^{V}(h_{0}),\phi\right\rangle,\,\text{
as long as }h_{n}\to h,$

by combining Corollary 3.7 and Lipschitz continuity of $h\to
P_{0,k,h}^{V}\phi(w)$ modulo the Lyapunov function $\mathfrak{m}_{\eta}$ and
growth in $k$.

To be specific, define

$\gamma_{k}^{\varphi}(h_{1},h_{2})=\left\langle
P_{0,k,\beta_{-k}h_{1}}^{V*}\Gamma^{V}(\beta_{-k}h_{2}),\varphi\right\rangle,\,\varphi\in
C_{b}(H),h_{1},h_{2}\in\mathbb{T}.$

Note that by Proposition 3.1, one has

$\gamma_{k}^{\mathbb{1}_{H}}(h,h)=\lambda_{k,\beta_{-k}h,V}\,\text{ and
}\,\gamma_{k}^{\phi}(h,h)=\gamma_{k}^{\mathbb{1}_{H}}(h,h)\left\langle\Gamma^{V}(h),\phi\right\rangle.$

Also observe that

(3.56)
$\displaystyle\begin{split}&\left|\left\langle\Gamma^{V}(h_{n}),\phi\right\rangle\
-\left\langle\Gamma^{V}(h_{0}),\phi\right\rangle\right|\\\
&\leq\left|\left\langle\Gamma^{V}(h_{n}),\phi\right\rangle\
-\frac{\gamma_{k}^{\phi}(h_{0},h_{n})}{\gamma_{k}^{\mathbb{1}_{H}}(h_{0},h_{n})}\right|+\left|\frac{\gamma_{k}^{\phi}(h_{0},h_{n})}{\gamma_{k}^{\mathbb{1}_{H}}(h_{0},h_{n})}-\left\langle\Gamma^{V}(h_{0}),\phi\right\rangle\right|:=I_{1,k}+I_{2,k}.\end{split}$

It follows from Corollary 3.7 that for any $\varepsilon>0$, there is a large
$K=K(h_{0})$ independent of $h_{n}$ such that for all $k\geq K$,

(3.57) $\displaystyle I_{2,k}<\varepsilon/2.$

Since $V\in C_{b}^{1,0}(H\times\mathbb{T})$, estimate as in (3.7) and the fact
$\Gamma^{V}(h)\in\mathcal{P}_{\eta,A}(H)$ yield the existence of some modulus
of continuity $\kappa$ and a constant $C=C(K,\phi,V)$ such that

$\displaystyle\left|\gamma_{K}^{\phi}(h_{n},h_{n})-\gamma_{K}^{\phi}(h_{0},h_{n})\right|$
$\displaystyle=\left|\left\langle
P_{0,K,\beta_{-K}h_{n}}^{V}\phi(w)-P_{0,K,\beta_{-K}h_{0}}^{V}\phi(w),\Gamma^{V}({\beta_{-K}h_{n}})\right\rangle\right|$
$\displaystyle\leq
C\left\langle\mathfrak{m}_{\eta},\Gamma^{V}(\beta_{-K}h_{n})\right\rangle(|h_{n}-h_{0}|+\kappa(|h_{n}-h_{0}|))$
$\displaystyle\leq C(|h_{n}-h_{0}|+\kappa(|h_{n}-h_{0}|)),$

for any Lipschitz $\phi\in C_{b}(H)$, where we used the fact that
$|\beta_{-K}h_{n}-\beta_{-K}h|=|h_{n}-h|$. Therefore by noting that
$e^{km_{V}}\leq\gamma_{k}^{\mathbb{1}_{H}}(h_{1},h_{2})\leq e^{kM_{V}}$ and
$|\gamma_{k}^{\phi}(h_{1},h_{2})|\leq\|\phi\|_{\infty}e^{kM_{V}}$ for any
$h_{1},h_{2}$, we have

$\displaystyle I_{1,K}\leq$
$\displaystyle\frac{\big{|}\gamma_{K}^{\mathbb{1}_{H}}(h_{0},h_{n})-\gamma_{K}^{\mathbb{1}_{H}}(h_{n},h_{n})\big{|}\big{|}\gamma_{K}^{\phi}(h_{n},h_{n})\big{|}}{\gamma_{K}^{\mathbb{1}_{H}}(h_{0},h_{n})\gamma_{K}^{\mathbb{1}_{H}}(h_{n},h_{n})}+\frac{\left|\gamma_{K}^{\mathbb{1}_{H}}(h_{n},h_{n})\right|\left|\gamma_{K}^{\phi}(h_{n},h_{n})-\gamma_{K}^{\phi}(h_{0},h_{n})\right|}{\gamma_{K}^{\mathbb{1}_{H}}(h_{0},h_{n})\gamma_{K}^{\mathbb{1}_{H}}(h_{n},h_{n})}$
$\displaystyle\leq C(K,\phi,V)(|h_{n}-h_{0}|+\kappa(|h_{n}-h_{0}|)).$

As a result, there is an $N=N(K,\varepsilon,\phi,V)$ such that for all $n\geq
N$,

(3.58) $\displaystyle I_{1,K}<\varepsilon/2.$

Hence $\Gamma^{V}\in C(\mathbb{T},\mathcal{P}_{\eta,A}(H))$ by combining
(3.56)-(3.58) and arbitrariness of bounded Lipschitz function $\phi$. The
continuity of $h\to\lambda_{k,h,V}$ follows from the fact that

$\displaystyle\lambda_{k,h,V}=\left\langle
P_{0,k,h}^{V}\mathbb{1}_{H},\Gamma^{V}(h)\right\rangle$

and continuity of $h\to\Gamma^{V}(h)$.

Step 3: Continuity of eigenfunction. The uniform Lipschitz continuity of the
eigenfunction family follows from the convergence (3.17) with
$\phi=\mathbb{1}_{H}$, the uniform Lipschitz property from Proposition C.4
with $\phi=\mathbb{1}_{H}$, and the uniform boundedness (3.14).

The proof of Proposition 3.8 is complete. ∎

###  3.4. Proof of Theorem 2.1

###### Proof of Theorem 2.1.

The proof is divided into three steps. We first show the uniqueness of eigen-
triple and then prove their continuity. Finally we prove the forward and
pullback stability properties.

Step 1: Uniqueness of eigen-triple for the continuous time family
$P^{V}_{0,t,h}$. Let $\lambda_{t,h,V}=P_{0,t,h}^{V*}\Gamma^{V}(h)(H)$. We
first show that $\Gamma^{V}$ is the unique eigenmeasureof the continuous time
Feynman-Kac evolution operators $P_{0,t,h}^{V}$ with eigenvalue
$\lambda_{t,h,V}$. For any $t_{0}>0$, we apply the process in the previous
three subsections to the discrete time Feynman-Kac evolution operators
$P_{0,kt_{0},h}^{V},k\geq 0$, which yields the existence of unique eigen-
triple $(\Gamma_{t_{0}}^{V},\widetilde{\lambda}_{kt_{0},h,V},F_{t_{0}}^{V})$.
Note that $\Gamma_{t_{0}}^{V}$ is the unique fixed point of the operator
$S_{t_{0}}^{V}$ as in (3.1), i.e.,

$S_{t_{0}}^{V}\Gamma_{t_{0}}^{V}=\Gamma_{t_{0}}^{V}.$

By the semigroup property of $S_{t}^{V}$, we have

$\displaystyle
S_{1}^{V}(S_{t_{0}}^{V}\Gamma^{V})=S_{t_{0}}^{V}(S_{1}^{V}\Gamma^{V})=S_{t_{0}}^{V}\Gamma^{V},$

implying that $S_{t_{0}}^{V}\Gamma^{V}$ is a fixed point of $S_{1}^{V}$, which
is unique by Proposition 3.8. Hence $S_{t_{0}}^{V}\Gamma^{V}=\Gamma^{V}$,
which in turn shows that $\Gamma^{V}$ is a fixed point of $S_{t_{0}}^{V}$.
Therefore by uniqueness

$\Gamma_{t_{0}}^{V}=\Gamma^{V},\quad\text{and
}\widetilde{\lambda}_{t_{0},h,V}=\lambda_{t_{0},h,V}.$

Since $t_{0}$ is arbitrary, we obtain that $\Gamma^{V}$, as the unique fixed
point of the semigroup $S_{t}^{V}$, is the unique eigenmeasure of
$P_{0,t,h}^{V}$ with eigenvalue $\lambda_{t,h,V}$.

The uniqueness of eigenfunction is similar. Indeed, let
$\widetilde{F}(w,h)=\lambda_{t_{0},h,V}^{-1}P_{0,t_{0},h}^{V}F^{V}(w,\beta_{t_{0}}h)$.
Then by using eigenfunction property of $F^{V}$ and cocycle property of
$\lambda_{t,h,V}$, one can verify in a straightforward manner that

$\left\langle\widetilde{F}(h),\Gamma^{V}(h)\right\rangle=1,\text{ and
}P_{0,k,h}^{V}\widetilde{F}(w,\beta_{k}h)=\lambda_{k,h,V}\widetilde{F}(w,h),\quad(w,h)\in
H\times\mathbb{T}.$

By uniqueness of eigenfunction from Proposition 3.8, one has
$\widetilde{F}=F^{V}$. Hence

$P_{0,t_{0},h}^{V}F^{V}(w,\beta_{t_{0}}h)=\lambda_{t_{0},h,V}F^{V}(w,h),$

which means that $F^{V}$ is an eigenfunction of the family
$\\{P_{0,kt_{0},h}^{V}\\}$. Therefore $F^{V}=F_{t_{0}}^{V}$ by uniqueness.

With the invariance (eigenmeasure property) of $\Gamma^{V}$ in the whole time
horizon, we are now able to show the integral representation (2.10) for the
eigenvalue through a differentiation procedure. Indeed, since $r\to H_{r}$ is
continuous almost surely, one has

$\displaystyle\frac{d}{dt}\lambda_{t,h,V}$
$\displaystyle=\frac{d}{dt}\int_{H}\mathbf{E}_{(w,h)}\left[e^{\int_{0}^{t}V(\mathcal{H}_{r})dr}\right]\Gamma^{V}(h)(dw)$
$\displaystyle=\int_{H}\mathbf{E}_{(w,h)}\left[e^{\int_{0}^{t}V(\mathcal{H}_{r})dr}V(H_{t})\right]\Gamma^{V}(h)(dw)$
$\displaystyle=\left\langle
V(\cdot,\beta_{t}h),P^{V*}_{0,t,h}\Gamma^{V}(h)\right\rangle=\lambda_{t,h,V}\left\langle
V(\cdot,\beta_{t}h),\Gamma^{V}(\beta_{t}h)\right\rangle.$

Since $\lambda_{0,h,V}=1$, by integrating the above equation we obtain the
desired representation

$\displaystyle\lambda_{t,h,V}=\exp\left(\int_{0}^{t}\lambda^{V}(\beta_{r}h)dr\right),\,\lambda^{V}(h)=\left\langle
V(\cdot,h),\Gamma^{V}(h)\right\rangle.$

The proof of Step 1 is complete.

Step 2: Continuity of the eigen-triple. It follows from Proposition 3.8 that
$\Gamma^{V}\in C(\mathbb{T},\mathcal{P}(H))$. The continuity of eigenvalue
follows from the fact that $V\in C_{b}^{1,0}(H\times\mathbb{T})$ and the
continuity of eigenmeasure once we note that

$\displaystyle\left|\lambda^{V}(h_{n})-\lambda^{V}(h)\right|$
$\displaystyle=\left|\left\langle
V(\cdot,h_{n}),\Gamma^{V}(h_{n})\right\rangle-\left\langle
V(\cdot,h),\Gamma^{V}(h)\right\rangle\right|$
$\displaystyle\leq\left|\left\langle
V(\cdot,h_{n}),\Gamma^{V}(h_{n})\right\rangle-\left\langle
V(\cdot,h),\Gamma^{V}(h_{n})\right\rangle\right|+\left|\left\langle
V(\cdot,h),\Gamma^{V}(h_{n})\right\rangle-\left\langle
V(\cdot,h),\Gamma^{V}(h)\right\rangle\right|$ $\displaystyle\leq
C\kappa(|h_{n}-h|)+\left|\left\langle
V(\cdot,h),\Gamma^{V}(h_{n})\right\rangle-\left\langle
V(\cdot,h),\Gamma^{V}(h)\right\rangle\right|\to 0,$

as long as $h_{n}\to h$, where $\kappa$ is some modulus of continuity coming
from the fact $V\in C_{b}^{1,0}(H\times\mathbb{T})$.

Since $\left\\{F^{V}(\cdot,h),h\in\mathbb{T}\right\\}$ is uniformly Lipschitz
on any bounded ball in $H$ by Proposition 3.8, we only need to show that $h\to
F^{V}(w,h)$ is continuous for every $w\in H$, which will imply that $F^{V}\in
C(H\times\mathbb{T})$. To that end, it suffices to show that

(3.59) $\displaystyle t\in\mathbb{R}\to F^{V}(w,\beta_{t}h)\,\text{ is
continuous for every }(w,h)\in H\times\mathbb{T}.$

Indeed, since $\mathbb{T}=[0,2\pi)$ and $\beta_{t}h=h+t\mod 2\pi$, we have
$h=\beta_{h}0$ for $h\in[0,2\pi)$. Hence

$\displaystyle h\to F^{V}(w,h)=F^{V}(w,\beta_{h}0),$

which is continuous by time continuity (3.59).

Therefore it remains to show (3.59). By the eigenfunction property, we have

(3.60) $\displaystyle
P_{0,t,\beta_{-t}h}^{V}F^{V}(\cdot,h)(w)=\lambda_{t,\beta_{-t}h,V}F^{V}(w,\beta_{-t}h),\,\forall
h\in\mathbb{T},t\geq 0.$

By the integral representation (2.10), we see that $\lambda_{t,\beta_{-t}h,V}$
is continuous in time. Hence we only need to show that
$P_{0,t,\beta_{-t}h}^{V}F^{V}(\cdot,h)(w)$ is time continuous.

Let $\tau\geq 0$, for notational simplicity denoting $F(w)=F^{V}(w,h),w\in H$
and

$\displaystyle\Xi_{t,h}^{V}:=\exp\left(\int_{0}^{t}V(\mathcal{H}_{r}(w,h))dr\right)=\exp\left(\int_{0}^{t}V(\Phi_{0,r,h}(w),\beta_{r}h)dr\right).$

Note that with $\mathbf{E}=\mathbf{E}_{w}$,

(3.61)
$\displaystyle\begin{split}&P_{0,t,\beta_{-t}h}^{V}F^{V}(\cdot,h)(w)-P_{0,\tau,\beta_{-\tau}h}^{V}F^{V}(\cdot,h)(w)\\\
&=\mathbf{E}\Xi_{t,\beta_{-t}h}^{V}F(\Phi_{0,t,\beta_{-t}h})-\mathbf{E}\Xi_{\tau,\beta_{-\tau}h}^{V}F(\Phi_{0,\tau,\beta_{-\tau}h})\\\
&=\mathbf{E}\left[(\Xi_{t,\beta_{-t}h}^{V}-\Xi_{\tau,\beta_{-\tau}h}^{V})F(\Phi_{0,t,\beta_{-t}h})\right]+\mathbf{E}\left[\Xi_{\tau,\beta_{-\tau}h}^{V}(F(\Phi_{0,t,\beta_{-t}h})-F(\Phi_{0,\tau,\beta_{-\tau}h}))\right]\\\
&:=I_{1}+I_{2}.\end{split}$

Since $F^{V}\in L^{\infty}_{\mathfrak{m}_{q}}(H\times\mathbb{T})$ and is
strictly positive everywhere by Proposition 3.2, one has

$\displaystyle|I_{1}|$
$\displaystyle\leq\mathbf{E}\left[\left|\Xi_{t,\beta_{-t}h}^{V}-\Xi_{\tau,\beta_{-t}h}^{V}\right|F(\Phi_{0,t,\beta_{-t}h})\right]+\mathbf{E}\left[\left|\Xi_{\tau,\beta_{-t}h}^{V}-\Xi_{\tau,\beta_{-\tau}h}^{V}\right|F(\Phi_{0,t,\beta_{-t}h})\right]$
$\displaystyle\leq
C\left(\mathbf{E}\left[\left|\int_{\tau}^{t}V(\mathcal{H}_{r}(w,\beta_{-t}h))dr\right|\mathfrak{m}_{q}\right]+\mathbf{E}\left[\int_{0}^{\tau}\left|V(\mathcal{H}_{r}(w,\beta_{-t}h))-V(\mathcal{H}_{r}(w,\beta_{-\tau}h))\right|dr\mathfrak{m}_{q}\right]\right)$
$\displaystyle\leq
C\left(|t-\tau|\mathbf{E}\mathfrak{m}_{q}+\kappa(|\beta_{-t}h-\beta_{-\tau}h|)\mathbf{E}\mathfrak{m}_{q}+\int_{0}^{\tau}\mathbf{E}\left[\mathfrak{m}_{q}\|\Phi_{0,r,\beta_{-t}h}-\Phi_{0,r,\beta_{-\tau}h}\|\right]dr\right)$
$\displaystyle\leq
C\left(|t-\tau|+\kappa(|\beta_{-t}h-\beta_{-\tau}h|)+|\beta_{-t}h-\beta_{-\tau}h|\right)$

for some modulus of continuity $\kappa$ coming from the fact $V\in
C_{b}^{1,0}(H\times\mathbb{T})$ and the constant $C=C(t,\tau,V)>0$. Here in
the second inequality we used the boundedness of $V,t,\tau$ and the Lipschitz
continuity of the exponential function $e^{x}$ on some bounded interval
determined by $m_{V},M_{V},t,\tau$. In the third inequality we used the fact
$V\in C_{b}^{1,0}(H\times\mathbb{T})$. In the last step we used Hölder’s
inequality and (B.6), and (B.4) together with the Lipschitz continuity of $f$.
Hence $I_{1}\to 0$ as $t\to\tau$.

Now observe that

$\displaystyle|I_{2}|$ $\displaystyle\leq
e^{\tau\|V\|_{\infty}}\mathbf{E}\left|F(\Phi_{0,t,\beta_{-t}h})-F(\Phi_{0,\tau,\beta_{-\tau}h})\right|$
$\displaystyle\leq
C\left(\mathbf{E}\left|F(\Phi_{0,t,\beta_{-t}h})-F(\Phi_{0,t,\beta_{-\tau}h})\right|+\mathbf{E}\left|F(\Phi_{0,t,\beta_{-\tau}h})-F(\Phi_{0,\tau,\beta_{-\tau}h})\right|\right)$
$\displaystyle:=C(I_{2,1}+I_{2,2}).$

Let
$\Omega_{R}=\left\\{\omega\in\Omega:\Phi_{0,t,\beta_{-t}h},\Phi_{0,t,\beta_{-\tau}h}\in
B_{R}^{H}\right\\}$. It follows from Proposition 3.8 that $F$ is Lipschitz
continuous on $B_{R}^{H}$ with some Lipschitz constant $L(R)$. Therefore by
$F^{V}\in L^{\infty}_{\mathfrak{m}_{q}}(H\times\mathbb{T})$, and (B.4)
together with the Lipschitz continuity of $f$, and noting that

$1\leq
R^{-2q}\left(\|\Phi_{0,t,\beta_{-t}h}\|^{2q}+\|\Phi_{0,t,\beta_{-\tau}h}\|^{2q}\right)\leq
R^{-2q}\left(\mathfrak{m}_{q}(\Phi_{0,t,\beta_{-t}h})+\mathfrak{m}_{q}(\Phi_{0,t,\beta_{-\tau}h})\right),\text{
on }\Omega_{R}^{c},$

one obtains

$\displaystyle I_{2,1}$
$\displaystyle=\mathbf{E}\left[\mathbb{1}_{\Omega_{R}}\left|F(\Phi_{0,t,\beta_{-t}h})-F(\Phi_{0,t,\beta_{-\tau}h})\right|\right]+\mathbf{E}\left[\mathbb{1}_{\Omega_{R}^{c}}\left|F(\Phi_{0,t,\beta_{-t}h})-F(\Phi_{0,t,\beta_{-\tau}h})\right|\right]$
$\displaystyle\leq
L(R)\mathbf{E}\left\|\Phi_{0,t,\beta_{-t}h}-\Phi_{0,t,\beta_{-\tau}h}\right\|+C\mathbf{E}\left[\mathbb{1}_{\Omega_{R}^{c}}\left(\mathfrak{m}_{q}(\Phi_{0,t,\beta_{-t}h})+\mathfrak{m}_{q}(\Phi_{0,t,\beta_{-\tau}h})\right)\right]$
$\displaystyle\leq
C\Big{(}L(R)|\beta_{-t}h-\beta_{-\tau}h|+R^{-2q}\mathbf{E}\left(\mathfrak{m}_{2q}(\Phi_{0,t,\beta_{-t}h})+\mathfrak{m}_{2q}(\Phi_{0,t,\beta_{-\tau}h})\right)\Big{)}$
$\displaystyle\leq
C\Big{(}L(R)|\beta_{-t}h-\beta_{-\tau}h|+R^{-2q}\mathfrak{m}_{2q}(w)\Big{)},$

where we used (B.6) at the last step. Therefore by first choosing $R$ large
and then letting $t\to\tau$, we have $I_{2,1}\to 0$.

Since each sample path of the solution $\Phi_{0,t,\beta_{-\tau}h},t\geq 0$ is
continuous in $t$, and $F^{V}\in
L^{\infty}_{\mathfrak{m}_{q}}(H\times\mathbb{T})$ implies

$\mathbf{E}\left|F(\Phi_{0,t,\beta_{-\tau}h})-F(\Phi_{0,\tau,\beta_{-\tau}h})\right|\leq\mathbf{E}\left(\mathfrak{m}_{q}(\Phi_{0,t,\beta_{-\tau}h})+\mathfrak{m}_{q}(\Phi_{0,\tau,\beta_{-\tau}h})\right)\leq
C\mathfrak{m}_{q}(w),\,t\geq 0,$

therefore by Lebesgue’s dominated convergence theorem, we have $I_{2,2}\to 0$
as $t\to\tau$. As a result $I_{2}\to 0$ as $t\to\tau$.

In view of (3.61) we obtain the time continuity of
$P_{0,t,\beta_{-t}h}^{V}F^{V}(\cdot,h)(w)$. Therefore by (3.60), the function

(3.62) $\displaystyle t\to
F^{V}(w,\beta_{-t}h)=\lambda_{t,\beta_{-t}h,V}^{-1}P_{0,t,\beta_{-t}h}^{V}F^{V}(\cdot,h)(w),\,t\geq
0$

is continuous for every $(w,h)\in H\times\mathbb{T}$. The continuity (3.59)
for any $t\in\mathbb{R}$ follows by shifting (replacing $h$ by $\beta_{s}h$
for any $s>0$ in (3.62)).

The proof of Step 2 is complete.

Step 3: Proof of the stability properties of Theorem 2.1. Let $g_{t}(w,h)$ be
the function obtained from (3.21) by replacing $k$ with $t$. The proof of the
forward convergence in this case is exactly the same as the proof of Theorem
3.3. Namely, we first show as in Proposition 3.5 the convergence of $g_{t}$
for $\phi\in C_{b}^{1}(H\times\mathbb{T})$ in $L^{1}(\Gamma^{V}(h))$ by
utilizing the unique eigen-triple and the local monotonicity (3.28) which is
also valid for continuous time. Then we proceed as in subsection 3.2.2.

The pullback convergence in $L^{1}(\Gamma^{V}(\beta_{-t}h))$ follows from the
convergence (3.46) in discrete time and the global monotonicity of
$t\to\left\langle|g_{t}(\cdot,\beta_{-t}h)|,\Gamma^{V}(\beta_{-t}h)\right\rangle$,
where the later follows by the semigroup property of $\mathcal{P}_{t}^{V}$,
and cocycle property of the eigenvalue, as well as the eigenmeasure property.
The other pullback stability properties in Theorem 2.1 then follow from
convergence in $L^{1}(\Gamma^{V}(\beta_{-t}h))$ and similar arguments as in
subsection 3.2.2.

The proof of Theorem 2.1 is complete.

∎

##  4\. A Donsker-Varadhan type large deviation principle

This section is devoted to the proof of Theorem 2.2. We prove this theorem in
the first subsection 4.1. Then in subsection 4.2 we prove several lemmas that
are used in the proof of Theorem 2.2.

###  4.1. Proof of Theorem 2.2

In this subsection, we prove the main result on the large deviation bounds in
Theorem 2.2 by combining the results of Theorem 2.1 and the Kifer type
criterion in Theorem A.1 from the Appendix, where the later was established in
[23].

More specifically, we first give a proof of Theorem 2.2 in subsection 4.1.1 by
assuming the following Theorem 4.1, which gives a verification of the
conditions in Kifer’s Theorem A.1. The proof of Theorem 4.1 is given in
subsection 4.1.2, which is based on Theorem 2.1.

######  Theorem 4.1.

Assume $A_{\infty}=H$ and $f\in C_{b}(\mathbb{R},H_{2})$ is Lipschitz. Let
$\eta\in(0,\eta_{0})$ and $M>0$.

  1. (1)

For any $V\in C_{b}(H\times\mathbb{T})$, the following limit called the
pressure function exists

(4.1) $\displaystyle
Q(V):=\lim_{t\to\infty}\frac{1}{t}\sup_{\sigma\in\mathcal{P}_{\eta,M}(H)}\log\mathbf{E}_{\sigma}\exp\left\langle
tV,\zeta_{t,h}\right\rangle,$

and independent of $h,\eta,M$. Also $Q(V)$ is a convex $1$-Lipschitz function
such that $Q(C)=C$ for any $C\in\mathbb{R}$. In particular for $V\in
C_{b}^{1,0}(H\times\mathbb{T})$,

(4.2) $\displaystyle Q(V)=\int_{\mathbb{T}}\lambda^{V}(h)m(dh),$

where $\lambda^{V}\in C(\mathbb{T},\mathbb{R})$ is the eigenvalue obtained in
Theorem 2.1.

  2. (2)

There is a function $R:H\times\mathbb{T}\to[0,\infty]$ with compact level sets
$\\{(w,h)\in H\times\mathbb{T}:R(w,h)\leq\alpha\\}$ for any $\alpha\geq 0$
such that

(4.3) $\displaystyle\mathbf{E}_{\sigma}\exp\left\langle
tR,\zeta_{t,h}\right\rangle\leq
Ce^{ct},\quad\sigma\in\mathcal{P}_{\eta,M}(H),$

for some positive constants $c,C$.

  3. (3)

For each $V\in C_{b}^{1,0}(H\times\mathbb{T})$, there is a unique equilibrium
state $\mu_{V}\in\mathcal{P}(H\times\mathbb{T})$, i.e.,

(4.4) $\displaystyle Q(V)=\langle V,\mu_{V}\rangle-I(\mu_{V}).$

Here $I$ is the Legendre transform of $Q$ as in (2.15).

####  4.1.1. Proof of Theorem 2.2

###### Proof.

The proof is divided into three steps.

Step 1: Link Theorem 4.1 to Theorem A.1. Note that the three conditions of
Kifer’s Theorem A.1 are verified by Theorem 4.1, where we take the directed
set

$\Theta=(0,\infty)\times\mathcal{P}_{\eta,M}(H),$

equipped with an order $\prec$ defined by
$(t_{1},\sigma_{1})\prec(t_{2},\sigma_{2})$ if $t_{1}\leq t_{2}$. And for each
fixed $h\in\mathbb{T}$, we set

$\displaystyle\zeta_{\theta}=\zeta_{t,h},\quad r(\theta)=t,\,\text{ for every
}\theta=(t,\sigma)\in\Theta,$

where $\zeta_{t,h}$ is the occupation measure defined as (2.12), which is a
family of random probability measures defined on the probability space
$(\Omega_{\theta},\mathcal{F}_{\theta},\mathbf{P}_{\theta})=(\Omega,\mathcal{F},\mathbf{P}_{\sigma})$.
Hence Theorem A.1 implies the desired large deviation bounds with the good
rate function

(4.7) $\displaystyle I(\mu)=\left\\{\begin{array}[]{cc}\displaystyle\sup_{V\in
C_{b}(H\times\mathbb{T})}\left(\left\langle
V,\mu\right\rangle-Q(V)\right),&\mu\in\mathcal{P}(H\times\mathbb{T}),\\\
\qquad+\infty,&\text{otherwise.}\end{array}\right.$

It remains to show that the good rate function is nontrivial.

Step 2: Prove the non-triviality of the rate function. Assume by contradiction
that $I(\gamma)=0$ and $I(\mu)=+\infty$ for
$\mu\in\mathcal{P}_{m}(H\times\mathbb{T})\setminus\\{\gamma\\}$, where
$\gamma=\Gamma(h)m(dh)$ is the unique invariant measure of the homogenized
process $\mathcal{H}_{t}$, and $\Gamma\in C(\mathbb{T},\mathcal{P}(H))$ is the
continuous periodic invariant measure of $\Phi_{0,t,h}$[28]. Note that
$\Gamma$ is also the unique eigenmeasure of $P_{0,t,h}^{V}$ in the case when
the potential $V=0$. Since $I$ is the Legendre transform of $Q$, it follows
from (4.7) that

$\displaystyle
Q(V)=\sup_{\mu\in\mathcal{P}(H\times\mathbb{T})}\left(\left\langle
V,\mu\right\rangle-I(\mu)\right),\quad V\in C_{b}(H\times\mathbb{T}).$

The assumption of triviality of $I$ implies that

(4.8) $\displaystyle Q(V)=\left\langle V,\gamma\right\rangle\text{ for any
}V\in C_{b}(H\times\mathbb{T}).$

To each $V\in C_{b}^{1}(H)$ we associate

$V_{\gamma}(w,h):=V(w)-\left\langle V,\Gamma(h)\right\rangle.$

It has been shown in [28] that $\Gamma\in C^{r}(\mathbb{T},\mathcal{P}(H))$
for some $r\in(0,1)$, where $\mathcal{P}(H)$ is equipped with a Wasserstein
metric weighted by the Lyapunov function $\mathfrak{m}_{\eta}$. Hence
$V_{\gamma}\in C_{b}^{1,0}(H\times\mathbb{T})$ for each $V\in C_{b}^{1}(H)$.
Since $\left\langle V_{\gamma},\gamma\right\rangle=0$, by (4.2) and (4.8), we
have

(4.9) $\displaystyle\int_{\mathbb{T}}\lambda^{V_{\gamma}}(h)m(dh)=0,\,V\in
C_{b}^{1}(H).$

Combining this with the integral representation (2.10) and Birkhoff’s ergodic
theorem, we have

(4.10)
$\displaystyle\lim_{t\to\infty}\frac{1}{t}\log\lambda_{t,h,V_{\gamma}}=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\lambda^{V_{\gamma}}(\beta_{rh})dr=\int_{\mathbb{T}}\lambda^{V_{\gamma}}(h)m(dh)=0,$

which implies that $\lambda_{t,h,V_{\gamma}}$ has a subexponential growth in
$t$.

The forward convergence (2.11) in Theorem 2.1 with $\phi=\mathbb{1}_{H}$ and
$\mu=\delta_{w}$ implies that

(4.11)
$\displaystyle\lim_{t\to\infty}\lambda_{t,h,V_{\gamma}}^{-1}P^{V_{\gamma}}_{0,t,h}\mathbb{1}_{H}(w)=F^{V_{\gamma}}(w,h),\,w\in
H,$

where we note that

(4.12) $\displaystyle
P^{V_{\gamma}}_{0,t,h}\mathbb{1}_{H}(w)=\mathbf{E}_{w}\exp\left(\int_{0}^{t}\big{(}V(\Phi_{0,r,h})-\left\langle
V,\Gamma(\beta_{r}h)\right\rangle\big{)}dr\right).$

On the other hand, the central limit theorem proved in [28] shows that

$\displaystyle\frac{1}{\sqrt{t}}\int_{0}^{t}\big{(}V(\Phi_{0,r,h})-\left\langle
V,\Gamma(\beta_{r}h)\right\rangle\big{)}dr\text{ is asymptotically Gaussian
with variance }\sigma_{V}^{2}.$

Hence if one can show the existence of $V\in C_{b}^{1}(H)$ such that
$\sigma_{V}^{2}\neq 0$, then (4.12) will imply that
$P^{V_{\gamma}}_{0,t,h}\mathbb{1}_{H}(w)$ grows exponentially fast in $t$,
which will contradicts with the subexponential growth (4.10) and the
convergence (4.11). This contradiction then shows that the rate function $I$
is nontrivial.

Step 3: Prove the existence of an observable function with nonzero asymptotic
variance. The rest of the proof is devoted to show the existence of such a
function $V$ that makes $\sigma_{V}^{2}\neq 0$ through a contradiction
argument as in [26] Proposition 4.1.4. For any $R>0$, we consider a
$C_{b}^{1}$ function $V_{R}:H\to\mathbb{R}$ defined as follows: $0\leq
V_{R}\leq 1$ and

(4.13) $\displaystyle V_{R}(w)=1,\,\text{ when }w\in B_{R}^{H},\quad\text{ and
}V_{R}(w)=0,\,\text{ when }w\in H\setminus B_{R+1}^{H}.$

Since $\Gamma$ is the unique eigenmeasure corresponding to $V=0$, it also has
the support property as in Proposition 3.1. Hence there is a $0<p=p(R)<1$ does
not depend on $h\in\mathbb{T}$ such that

$\displaystyle\Gamma(h)(B_{R+1}^{H})\leq p<1,\,h\in\mathbb{T},$

which implies $\left\langle V_{R},\Gamma(h)\right\rangle\leq p$. As a result,
one has

(4.14) $\displaystyle V_{\gamma,R}(w,h):=V_{R}(w)-\left\langle
V_{R},\Gamma(h)\right\rangle\geq 1-p>0,\,\forall w\in
B_{R}^{H},h\in\mathbb{T}.$

Assume by contradiction that $\sigma_{V_{\gamma,R}}^{2}=0$. Recall the
martingale approximation as in [28],

(4.15)
$\displaystyle\int_{0}^{N}V_{\gamma,R}(\mathcal{H}_{r})dr=M_{N}-\chi(\mathcal{H}_{N})+\chi(\mathcal{H}_{0}),$

where
$M_{N}=\int_{0}^{N}V_{\gamma,R}(\mathcal{H}_{r})dr+\chi(\mathcal{H}_{N})-\chi(\mathcal{H}_{0})$
is the Dynkin martingale and the corrector $\chi$ is defined as (Note that
$\mathcal{P}_{t}=\mathcal{P}_{t}^{V}$ with $V=0$)

$\chi(w,h)=\int_{0}^{\infty}\mathcal{P}_{t}V_{\gamma,R}(w,h)dr,\,(w,h)\in
H\times\mathbb{T},$

which is bounded modulo $e^{\eta\|w\|^{2}}$, see [28]. We claim that the
assumption $\sigma_{V_{\gamma,R}}^{2}=0$ implies

(4.16) $\displaystyle\mathbf{P}_{\gamma}\left\\{M_{N}=0,\,\forall N\geq
0\right\\}=1.$

Indeed, by boundedness of $\chi$ and integrability of $e^{\eta\|w\|^{2}}$, we
have

$\displaystyle\mathbf{E}_{\gamma}M_{N}^{2}$ $\displaystyle\leq
3\mathbf{E}_{\gamma}\left(\int_{0}^{N}V_{\gamma,R}(\mathcal{H}_{r})dr\right)^{2}+3\mathbf{E}_{\gamma}|\chi(\mathcal{H}_{N})|+3\mathbf{E}_{\gamma}|\chi(\mathcal{H}_{0})|$
$\displaystyle\leq
C\left(\mathbf{E}_{\gamma}\left(\int_{0}^{N}V_{\gamma,R}(\mathcal{H}_{r})dr\right)^{2}+1\right).$

Therefore by Proposition 4.12 in [28] on the asymptotic variance
$\sigma_{V_{\gamma,R}}^{2}$, we have

(4.17)
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\mathbf{E}_{\gamma}M_{N}^{2}=0,\text{
since
}\lim_{N\to\infty}\frac{1}{N}\mathbf{E}_{\gamma}\left(\int_{0}^{N}V_{\gamma,R}(\mathcal{H}_{r})dr\right)^{2}=\sigma_{V_{\gamma,R}}^{2}=0.$

Since $\gamma$ is the invariant measure of the homogenized process, for the
martingale difference sequence $Z_{k}=M_{k}-M_{k-1}$, one has
$\mathbf{E}_{\gamma}Z_{k}^{2}=\mathbf{E}_{\gamma}Z_{1}^{2}$ for all $k\geq 1$
and

$\displaystyle\mathbf{E}_{\gamma}M_{N}^{2}=\mathbf{E}_{\gamma}\left(\sum_{k=1}^{N}Z_{k}\right)^{2}=\mathbf{E}_{\gamma}\sum_{k=1}^{N}Z_{k}^{2}=N\mathbf{E}_{\gamma}Z_{1}^{2}.$

Combining this with (4.17) we obtain $\mathbf{E}_{\gamma}Z_{k}^{2}=0$ for all
$k$. Hence $M_{N}=0$ almost surely for all $N$ and the claim (4.16) is proved.

We are now ready to derive a contradiction. Note that by deterministic version
of the estimate (B.6), there is a constant $C>1$ independent of
$h\in\mathbb{T}$ such that the deterministic solution $\Phi_{0,t,h}^{d}(w)$ to
equation (2.2) without noise satisfies (see also [8])

$\displaystyle\|\Phi_{0,t,h}^{d}(w)\|^{2}\leq e^{-\nu t}\|w\|^{2}+C,\,\forall
w\in H,h\in\mathbb{T}.$

Hence by choosing $R=2C$ in the definition (4.13) of $V_{R}$, we have that for
any initial condition $(w,h)\in B_{1}^{H}\times\mathbb{T}$, the deterministic
solution $\Phi_{0,t,h}^{d}(w)\in B_{R}^{H}$ for all $t\geq 0$. Therefore for
any initial $(w,h)\in B_{1}^{H}\times\mathbb{T}$ and $N\geq 0$, there is a
small positive probability (corresponding to those Brownian sample paths in a
small neighborhood of the origin), such that the solution $\Phi_{0,t,h}(w)$ to
the noisy system (2.2) always stays in $B_{R}^{H}$ for $t\in[0,N]$. Namely, we
have

$\displaystyle\mathbf{P}_{(w,h)}\left\\{\mathcal{H}_{r}\in
B_{R}^{H}\times\mathbb{T},\text{ for }0\leq r\leq N\right\\}>0,\,(w,h)\in
B_{1}^{H}\times\mathbb{T}.$

Since $\gamma(B_{1}^{H}\times\mathbb{T})>0$, one has

$\displaystyle\mathbf{P}_{\gamma}\left\\{\mathcal{H}_{r}\in
B_{R}^{H}\times\mathbb{T},\text{ for }0\leq r\leq N\right\\}>0.$

In view of (4.14), we obtain for any $N\geq 0$,

$\displaystyle\mathbf{P}_{\gamma}\left\\{\int_{0}^{N}V_{\gamma,R}(\mathcal{H}_{r})dr\geq
N(1-p)\right\\}>0.$

This combined with (4.16), and boundedness of the corrector $\chi$ over
$B_{R}^{H}\times\mathbb{T}$, imply that (4.15) cannot hold with
$\mathbf{P}_{\gamma}$ probability $1$. This contradiction shows that
$\sigma_{V_{\gamma,R}}^{2}\neq 0$.

The proof is complete. ∎

####  4.1.2. Proof of Theorem 4.1

We now supply a proof of Theorem 4.1 based on the results of Theorem 2.1.

Proof of Items (1) and (2) in Theorem 4.1. Note that Item (2) in Theorem 4.1
directly follows from (B.2) with $R(w,h)=\eta\nu\|w\|_{1}^{2}$. The level set
$\\{(w,h)\in H\times\mathbb{T}:R(w,h)\leq\alpha\\}$ for any $\alpha\geq 0$ is
compact since $\mathbb{T}$ is compact and the embedding $H_{1}\to H$ is
compact.

Item (1) of Theorem 4.1 follows from the forward convergence in Theorem 2.1,
together with Birkhoff’s ergodic theorem and a $buc$-approximation argument.
Indeed, taking $\phi=\mathbb{1}_{H}$ in the forward convergence (2.11) in
Theorem 2.1, and noting the strict positivity of eigenfunction $F^{V}$, we
have for $V\in C_{b}^{1,0}(H\times\mathbb{T})$,

(4.18)
$\displaystyle\lim_{t\to\infty}\left|\frac{1}{t}\log\lambda_{t,h,V}-\frac{1}{t}\log\left\langle
P_{0,t,h}^{V}\mathbb{1}_{H},\mu\right\rangle\right|=0,$

uniformly for $\mu\in\mathcal{P}_{\eta,M}(H)$. The integral representation
(2.10) and Birkhoff’s ergodic theorem, as well as the continuity of
$h\to\lambda^{V}(h)$ imply that

(4.19)
$\displaystyle\lim_{t\to\infty}\frac{1}{t}\log\lambda_{t,h,V}=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\lambda^{V}(\beta_{r}h)dr=\int_{\mathbb{T}}\lambda^{V}(h)m(dh).$

A combination of (4.18) and (4.19) verifies (4.1) for $V\in
C_{b}^{1,0}(H\times\mathbb{T})$ and (4.2) in Item (1) of Theorem 4.1. The
existence of the limit in (4.1) for general $V\in C_{b}(H\times\mathbb{T})$
follows from a $buc$-approximation argument as in [23], where the required
exponential tightness has been verified in Item (2). The Lipschitz continuity
is derived from the limit definition of $Q$. Indeed, note that for any
$\sigma\in\mathcal{P}_{\eta,M}(H)$,

$\displaystyle\mathbf{E}_{\sigma}\exp\left\langle
tV_{1},\zeta_{t,h}\right\rangle$
$\displaystyle=\mathbf{E}_{\sigma}\exp\big{(}\left\langle
t(V_{1}-V_{2}),\zeta_{t,h}\right\rangle+\left\langle
tV_{2},\zeta_{t,h}\right\rangle\big{)}$ $\displaystyle\leq
e^{t\|V_{1}-V_{2}\|_{\infty}}\mathbf{E}_{\sigma}\exp\left\langle
tV_{2},\zeta_{t,h}\right\rangle.$

By exchanging the role of $V_{1},V_{2}$, we have

$\displaystyle\left|\frac{1}{t}\log\mathbf{E}_{\sigma}\exp\left\langle
tV_{1},\zeta_{t,h}\right\rangle-\frac{1}{t}\log\mathbf{E}_{\sigma}\exp\left\langle
tV_{2},\zeta_{t,h}\right\rangle\right|\leq\|V_{1}-V_{2}\|_{\infty},$

which by taking limit, implies the Lipschitz continuity of $Q$. The convexity
follows from Hölder’s inequality. The proof of Item (1) is then complete.

Proof of Item (3) in Theorem 4.1. The remainder of this subsection is then
devoted to prove Item (3) on uniqueness of equilibrium state in Theorem 4.1,
which is accomplished through the following three steps. Fix any $V\in
C_{b}^{1,0}(H\times\mathbb{T})$.

Step 1: The auxiliary Markov evolution operators defined through Doob’s
$h$-transform is uniquely ergodic. Define the following auxiliary evolution
family of Markov operators $T^{V}_{0,t,h}$ acting on $C_{b}(H)$,

(4.20) $\displaystyle
T^{V}_{0,t,h}\phi(w)=\lambda_{t,h,V}^{-1}F^{V}(w,h)^{-1}P_{0,t,h}^{V}(\phi
F^{V}(\cdot,\beta_{t}h))(w),\,\text{ for }(w,h)\in H\times\mathbb{T},\phi\in
C_{b}(H).$

This is an analogue of the Doob’s $h$-transform in our current setting. See
[4] and references therein for an introduction about the Doob’s $h$-transform,
and [23, 30] for the application in proving the uniqueness of equilibrium
state in the time homogeneous case.

The operator $T^{V}_{0,t,h}$ is Markov since
$T^{V}_{0,t,h}\mathbb{1}_{H}=\mathbb{1}_{H}$ by the eigenfunction property of
$F^{V}$. It follows from the integral representation (2.10) that the
eigenvalue satisfies the cocycle property

$\lambda_{t+s,h,V}=\lambda_{t,\beta_{s}h,V}\lambda_{s,h,V},\,\text{ for
}s,t\geq 0,h\in\mathbb{T}.$

Base on this and the cocycle property (2.7) of the Feynman-Kac operators, it