# Exponential mixing and limit theorems of quasi-periodically forced 2D
stochastic Navier-Stokes Equations in the hypoelliptic setting

Rongchang Liu  Department of Mathematics
Brigham Young University
Provo, Utah 84602, USA<EMAIL_ADDRESS>and  Kening Lu
Department of Mathematics
Brigham Young University
Provo, Utah 84602, USA<EMAIL_ADDRESS>
###### Abstract.

We consider the incompressible 2D Navier-Stokes equations on the torus driven
by a deterministic time quasi-periodic force and a noise that is white in time
and extremely degenerate in Fourier space. We show that the asymptotic
statistical behavior is characterized by a uniquely ergodic and exponentially
mixing quasi-periodic invariant measure. The result is true for any value of
the viscosity $\nu>0$ and does not depend on the strength of the external
forces, which in particular solves a conjecture made by Hairer and Mattingly
[24].

By utilizing this quasi-periodic invariant measure, we are able show the
strong law of large numbers and central limit theorem for the continuous time
inhomogeneous solution processes. Estimates of the corresponding rate of
convergence are also obtained, which is the same as in the time homogeneous
case for the strong law of large numbers, while the convergence rate in the
central limit theorem depends on the Diophantine approximation property on the
quasi-periodic frequency and the mixing rate of the quasi-periodic invariant
measure. We also prove the existence of a stable quasi-periodic solution in
the laminar case (when the viscosity is large).

The scheme of analyzing the statistical behavior of the time inhomogeneous
solution process by the quasi-periodic invariant measure is new and could be
extended to other inhomogeneous Markov processes.

###### Key words and phrases:

Navier-Stokes Equations, quasi-periodic invariant measure, unique ergodicity,
mixing, limit theorems, rate of convergence, Diophantine condition.

###### 2000 Mathematics Subject Classification:

Primary: 37D45, 37C40

###### Contents

  1. 1 Introduction
    1. 1.1 Description of results
    2. 1.2 Relation to existing works
  2. 2 Settings and main results
    1. 2.1 Well-posedness, quasi-periodic invariant measures and quasi-periodic solutions
    2. 2.2 Homogenization through skew product
    3. 2.3 Main results
  3. 3 Ergodicity and exponentially mixing
    1. 3.1 Contraction on $\mathcal{P}(H)$
      1. 3.1.1 Irreducibility: The Proof of Proposition 3.3
    2. 3.2 Ergodicity and mixing
  4. 4 SLLN and CLT for time inhomogeneous solution process
    1. 4.1 The martingale approximation
    2. 4.2 The limit theorems
    3. 4.3 The rate of convergence in the limit theorems
      1. 4.3.1 The rate of convergence in SLLN
      2. 4.3.2 The rate of convergence in CLT
  5. 5 Large viscosity implies a trivial quasi-periodic attractor
  6. 6 Appendix: Estimates of the solution

##  1\. Introduction

We study the asymptotic statistical properties of time inhomogeneous solution
process of the incompressible 2D Navier-Stokes equation driven by a
deterministic time-dependent quasi-periodic force and a highly degenerate
stochastic force. Besides its own interest in the theory of turbulence, the
consideration of the system leads to an effective scheme of analyzing the
statistical dynamics of time inhomogeneous Markov processes, which is known to
be challenging in the literature. More specifically, there are three main
ingredients of our work for this system:

(1) We extend the Harris-like theorem in the infinite dimensional hypoelliptic
setting developed by Hairer and Mattingly [24] to the time quasi-periodic
inhomogeneous case. Furthermore, by viewing the two-parameter Markov
transition operator as a contractive deterministic non-autonomous dynamical
system acting on the space of probability measures, we show that the non-
autonomous system has a globally exponentially attracting quasi-periodic
trajectory, serving as the uniquely ergodic and exponentially mixing quasi-
periodic invariant measure, which describes the quasi-periodically statistical
steady states of the system.

(2) The quasi-periodic invariant measure in (1) enables us to develop a
martingale approximation scheme, to show the strong law of large numbers and
central limit theorem for the time inhomogeneous solution process starting at
every deterministic point and for Hölder observables weighted by a Lyapunov
function. This particular type of martingale approximation as well as the
limit theorems for the time inhomogeneous Markov solution process, are new in
the literature. They reflect how the asymptotic statistical dynamics of the
time inhomogeneous solution process is characterized by the quasi-periodic
invariant measure.

(3) We also obtain an estimate on the rate of convergence in the limit
theorems, which is apparently new in the time inhomogeneous regime. For the
strong law of large numbers, the convergence rate is the same as in the time
homogeneous case. While in the case of the central limit theorem, the
convergence rate is related to the mixing rate of the quasi-periodic invariant
measure and the convergence rate of the Birkhoff average of the irrational
rotation flow with particular observable functions involving the quasi-
periodic invariant measure. The later has a close connection with the
Diophantine approximation property of the frequency. The result reveals how
the statistical dynamics of a time inhomogeneous Markov process is affected by
its time inhomogeneity.

We confine ourselves to the 2D Navier-Stokes system to simplify the
presentation of the main ideas. However, the results can be applied to a class
of time inhomogeneous Markov process generated by stochastic semilinear
equations driven by an additional time dependent deterministic force for which
the in-homogeneity can be modeled by a dynamical system (the Bebutov shift for
example). In what follows we will briefly describe our results (Theorems A, B
and C below) and compare them with the existing literature, while leave the
technical preliminaries to the next section.

###  1.1. Description of results

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

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

where $w(t,x)$ is the vorticity field, and $\mathcal{K}w$ is the divergence
free velocity field. 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 $\|\cdot\|$. The deterministic force $f$ is quasi-
periodic in $t$ in the sense that $f(t,x)=\Psi(\alpha t,x)$ for some $\Psi\in
C(\mathbb{T}^{n},H)$. Here the frequency
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$ and
$\\{\alpha_{k}\\}_{k=1}^{n}$ are rationally independent.
$W=(W_{1},W_{2},\cdots,W_{d})$ 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 $\\{g_{i}\\}$ are elements of $H$.
Under appropriate spatial regularity conditions on the external forces (see
the next section), the equation is well posed with a time inhomogeneous Markov
solution process $w_{s,t}(w_{0})$. It generates a two-parameter Markov
transition operator $\mathcal{P}_{s,t}$ acting on the space of bounded
measurable functions ${B}_{b}(H)$ as

$\mathcal{P}_{s,t}\phi(w_{0})=\mathbf{E}\phi(w_{s,t}(w_{0})),\quad\forall\varphi\in
B_{b}(H),w_{0}\in H.$

It acts on the space of probability measures $\mathcal{P}(H)$ by duality

$\mathcal{P}_{s,t}^{*}\mu(A)=\int_{H}\mathcal{P}_{s,t}\mathbb{I}_{A}(w)\mu(dw),\text{
for }\mu\in\mathcal{P}(H),A\in\mathfrak{B},$

where $\mathfrak{B}$ is the Borel $\sigma$-algebra of $H$ and $\mathbb{I}_{A}$
is the indicator function of $A$. For $\eta>0$ small, recall the geodesic
metric $\rho$ weighted by a Lyapunov function introduced in [24]

(1.1)
$\displaystyle\rho(w_{1},w_{2})=\inf_{\gamma}\int_{0}^{1}e^{\eta\|\gamma(t)\|^{2}}\|\dot{\gamma}(t)\|dt,\quad\forall
w_{1},w_{2}\in H,$

where the infimum is taken over all differentiable path $\gamma$ connecting
$w_{1},w_{2}\in H$. We endow $\mathcal{P}(H)$ with the topology of weak
convergence and denote by $\mathcal{P}_{1}(H)$ the set of probability measures
that have finite first moment with respect to the 1-Wasserstein metric induced
from the metric $\rho$ in $H$. A quasi-periodic invariant measure is a
continuous quasi-periodic map with values in $\mathcal{P}(H)$ satisfying the
following invariance condition

(1.2) $\displaystyle\mathcal{P}_{s,t}^{*}\mu_{s}=\mu_{t},\quad s\leq t.$

To state the main result, we recall a condition on the structure of the
degenerate noise from [25]. Define the set $A_{\infty}$ by setting
$A_{1}=\\{g_{l}:1\leq l\leq d\\}$,
$A_{k+1}=A_{k}\cup\\{\widetilde{B}(h,g_{l}):h\in A_{k},g_{l}\in A_{1}\\}$, and
$A_{\infty}=\overline{\mathrm{span}(\cup_{k\geq 1}A_{k})}$, where
$\widetilde{B}(u,w)=-B(\mathcal{K}u,w)-B(\mathcal{K}w,u)$ is the symmetrized
nonlinear term. These sets reflect the mechanism of the propagation of the
extremely degenerate noise to the phase space that yields a smoothing effect
of the dynamics.

Besides spatial regularity conditions on the external forces, the only
remaining assumption for our main results is the Hörmander condition:
$A_{\infty}=H$. In particular, the noise is allowed to be extremely degenerate
to have $A_{\infty}=H$, for example it can be excited only through four
directions [23].

######  Remark 1.1.

Note that our result does not rely on any condition on the viscosity $\nu>0$,
nor conditions on the strength of the external forces. In particular, we do
not need the range condition as in [24] and our result in the case when $f$ is
time independent verifies a conjecture made by Hairer and Mattingly [24] (see
Remark 1.3) that the spectral gap (as well as unique ergodicity and
exponentially mixing) holds without restrictions on $f$ other than it be
sufficiently smooth.

Unique ergodicity and mixing of the quasi-periodic invariant measure

One of the main ingredients of this paper is to provide a perspective to
investigate the asymptotic statistical behaviors of the time inhomogeneous
Markov solution process, by analyzing the dynamics of the non-autonomous
system on $\mathcal{P}(H)$ induced by the action of the two parameter Markov
transition operators $\mathcal{P}_{s,t}^{*}$. It turns out that the asymptotic
behavior of $\mathcal{P}_{s,t}^{*}$ is characterized by a complete quasi-
periodic trajectory in $\mathcal{P}(H)$. More specifically, we prove

######  Theorem A (Exponential Mixing).

There exists a unique quasi-periodic path $\\{\mu_{t}\\}_{t\in\mathbb{R}}$ in
$\mathcal{P}(H)$ satisfying the invariance condition (1.2), such that

(1.3) $\displaystyle\rho(\mathcal{P}_{s,t}^{*}\mu,\mu_{t})\leq
Ce^{-\varpi(t-s)}\rho(\mu,\mu_{s}),\quad\forall s\leq t,\mu\in\mathcal{P}(H),$

where $C,\varpi$ are positive constants.

Such a unique quasi-periodic path is called a uniquely ergodic and
exponentially mixing quasi-periodic invariant measure. One of the classical
methods to show the unique ergodicity and mixing in the homogeneous setting,
which dates back to the early works of Doeblin [12] and Harris [21], is to
show that the action of the Markov semigroup on $\mathcal{P}(H)$ is a
contraction under an appropriate metric, whose unique fixed point gives the
uniquely ergodic and mixing invariant measure. The Harris work was generalized
to the hypoelliptic case for infinite dimensional systems by Hairer and
Mattingly [24]. Our first result (1.3) can be regarded as an extension of
their work to the time (quasi-periodically) inhomogeneous setting.

The proof of Theorem A will be given in section 3 by first proving
$\mathcal{P}_{s,t}^{*}$ is a contraction on $\mathcal{P}(H)$ and then using a
fixed point argument to the induced action (through a pull-back procedure) on
the space of quasi-periodic graphs $C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$.
Besides the Lyapunov structure and a particular type of irreducibility (in a
form that is more uniform than the usual topological irreducibility), the
proof of the contraction for $\mathcal{P}_{s,t}^{*}$ requires a deep analysis
of its gradient in the time inhomogeneous hypoelliptic setting, which was
first developed in the time homogeneous case by Hairer and Mattingly in their
celebrated works [23, 24, 25]. In addtion, we prove the particular type of
irreducibility here by combining the parabolic regularizing effect of the
equation and the usual topological irreducibility, where the later is a
consequence of the well known controllability results of Agrachev and Sarychev
[1, 2].

Limit theorems in terms of the quasi-periodic invariant measure

In the time homogeneous setting, the strong law of large numbers and central
limit theorem show that the asymptotic behavior of an observation along a
Markov process can be characterized by the unique invariant measure. Suppose
that we are given a homogeneous Markov process $X_{t}$ with a unique invariant
measure $\mu_{*}$ with a certain independence condition (mixing for example)
and $\phi\in C(H,\mathbb{R})$ is an observable function with some regularity,
say Hölder continuous. Then as $T\rightarrow\infty$, one has (see [28] for
example) the strong law of large numbers

(1.4)
$\displaystyle\frac{1}{T}\int_{0}^{T}\phi(X_{t})dt\overset{\mathrm{a.s.}}{\longrightarrow}\langle\mu_{*},\phi\rangle,$

and the central limt theorem

(1.5)
$\displaystyle\frac{1}{\sqrt{T}}\int_{0}^{T}\Big{(}\phi(X_{t})-\langle\mu_{*},\phi\rangle\Big{)}dt\overset{\mathcal{D}}{\longrightarrow}N(0,\sigma^{2}),$

where $\langle\mu_{*},\phi\rangle$ is the integral of $\phi$ with respect to
$\mu_{*}$, and $\mathrm{a.s.}$ denotes the almost sure convergence w.r.t. the
Wiener measure $\mathbf{P}$, while $\mathcal{D}$ represents the convergence in
distribution and $N(0,\sigma^{2})$ is the centered normal distribution with
variance $\sigma^{2}\geq 0$. The strong law of large numbers in the form (1.4)
also bears the name of the ergodic theorem, which states that the time average
of the observations converges to the ensemble average for almost every sample,
regardless of the initial condition. And the central limit theorem measures
the size of fluctuations around the ensemble average. We notice that the
following form of the strong law of large numbers

(1.6)
$\displaystyle\frac{1}{T}\int_{0}^{T}\Big{(}\phi(X_{t})-\langle\mu_{*},\phi\rangle\Big{)}dt\overset{\mathrm{a.s.}}{\longrightarrow}0$

captures another role played by the invariant measure in the sense that, the
average of observations centered by the invariant measure converges to the
corresponding asymptotic mean. We will see that the quasi-periodic invariant
measure has the same feature, which is the key to extend the limit theorems to
the time inhomogeneous setting.

The celebrated Dobrusion’s theorem [11] is of particular importance in the
time inhomogeneous case. It shows that for any discrete time inhomogeneous
Markov chain $X_{k}$ with a certain compatibility condition between the
minimal ergodic coefficients, the observable functions $\phi_{k}$ and the
variance, one has the convergence as $n\rightarrow\infty$,

(1.7)
$\displaystyle\frac{S_{n}-\mathbf{E}[S_{n}]}{\sqrt{\mathrm{Var}(S_{n})}}\overset{\mathcal{D}}{\longrightarrow}N(0,1),$

where $\displaystyle S_{n}=\sum_{k=1}^{n}\phi_{k}(X_{k})$ and
$\mathrm{Var}(S_{n})$ is the variance. Although this theorem is quite general
in its own right, it is not applicable to the Navier-Stokes equation in our
context. Indeed, as mentioned in [24], the transition probabilities in
infinite dimensional systems are likely to be mutually singular, especially in
the case when the strong Feller property does not hold. Hence the non-
degeneracy condition (characterized by the total variational metric) on the
minimal ergodic coefficients in Dobrusion’s theorem may not be satisfied.
Besides, to apply the Dobrusion’s theorem, one needs to compute the
expectation and variance along each observation as indicated in (1.7).

Our second result shows how the uniquely ergodic and mixing quasi-periodic
invariant measure $\mu_{t}$ enables us to give the limit theorems for the time
inhomogeneous solution process. Indeed, in view of the role played by the
invariant measure in the homogeneous case, and the fact that the ergodicity
and mixing of the quasi-periodic invariant measure shows that the distribution
of any solution is exponentially attracted by the quasi-periodic path in
$\mathcal{P}(H)$, one may expect that the quasi-periodic invariant measure
carries the information that one needs to center the observations
appropriately to derive the limit theorems. In fact, we obtain the following
limit theorems.

######  Theorem B (Limit Theorems).

For any Hölder continuous observable function $\phi$, and any initial data
$w_{0}\in H$, the solution $w_{s,s+t}(w_{0})$ of the Navier-Stokes system
satisfies as $T\rightarrow\infty$ the strong law of large numbers

(1.8)
$\displaystyle\frac{1}{T}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle\Big{)}dt\overset{\mathrm{a.s.}}{\longrightarrow}0,$

and the central limit theorem

(1.9)
$\displaystyle\frac{1}{\sqrt{T}}\int_{0}^{T}\Big{(}\phi\left(w_{s,s+t}(w_{0})\right)-\left\langle\mu_{s+t},\phi\right\rangle\Big{)}dt\overset{\mathcal{D}}{\longrightarrow}N(0,\sigma^{2}).$

Theorem B are proved in section 4 through a martingale approximation procedure
and by applying martingale limit theorems. The idea of the martingale
approximation was originally due to Gordin [20] and further developed by Knips
and Varadhan [27]. In [35], a new proof of the Dobrusion’ s theorem was given
by using similar ideas. While our approach here is different from that in
[35], since we apply the martingale approximation to the associated
homogenized Markov process on the extended phase space
$H\times\mathbb{T}^{n}$, where $\mathbb{T}^{n}$ is equipped with the
irrational rotation flow with rotation frequency $\alpha$ from the quasi-
periodic force (see subsection 2.2 below). It is worth to mention that this
homogenized process is not mixing since the irrational rotation flow is never
mixing. Hence the method of martingale approximation for uniformly mixing time
homogeneous Markov processes [28] cannot be applied in a straightforward way.
Fortunately, the exponentially mixing quasi-periodic invariant measure centers
the observations in the limit theorems in an appropriate way that enables us
to derive a martingale approximation.

Rate of convergence in the limit theorems In the time homogeneous and
essentially elliptic setting, the estimates of the rate of convergence of the
limit theorems were obtained for the 2D stochastic Navier-Stokes equation,
which are close to be optimal [34]. Our third result shows that similar
estimates hold in the inhomogeneous context. Namely, we will show the
following convergence rate (which is the same as in the time homogeneous case)
for the limit theorems:

######  Theorem C (Convergence Rates).

1\. Convergence rate of the strong law of large numbers: for any
$s\in\mathbb{R}$ and $\varepsilon>0$, there is an almost surely finite random
time $T_{s,\varepsilon}$ such that for all $T\geq T_{s,\varepsilon}$,

(1.10)
$\displaystyle\left|\frac{1}{T}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle\Big{)}dt\right|\leq
CT^{-\frac{1}{2}+\varepsilon},$

2\. Convergence rate of the central limit theorem:

(1.11)
$\displaystyle\sup_{z\in\mathbb{R}}\left(\xi_{\sigma}(z)\left|\mathbf{P}\left\\{\frac{1}{\sqrt{T}}\int_{0}^{T}\Big{(}\phi\left(w_{s,s+t}(w_{0})\right)-\left\langle\mu_{s+t},\phi\right\rangle\Big{)}dt\leq
z\right\\}-\Phi_{\sigma}(z)\right|\right)\leq
C_{\varepsilon_{0}}T^{-\frac{1}{4}+\varepsilon_{0}},$

where $\xi_{\sigma}\equiv 1$ for $\sigma>0,\xi_{0}(z)=1\wedge|z|$, and
$\Phi_{\sigma}(z)$ is the distribution function of the centered Gaussian
distribution.

Here $\varepsilon_{0}\in(0,\frac{1}{4})$ is a constant depending on the mixing
rate of the quasi-periodic invariant measure and the convergence rate for the
Birkhoff ergodic sums of the irrational rotation for a particular observable
function involving the quasi-periodic invariant measure.

Theorem C is proved in section 4. The general idea is to derive first the
estimates for the approximating martingale and then pass to inequalities
(1.10) and (1.11) by invoking the martingale approximation. In particular,
estimate (1.11) is derived from a combination of several ideas from [34] with
an important Berry-Esseen type result for martingales from [22]. We stress
here that estimate (1.11) requires a detailed analysis on the Hölder
regularity of a particular induced observable function (see Proposition 4.13)
on the torus $\mathbb{T}^{n}$ involving the quasi-periodic invariant measure.
This is a main feature in our context that is different from the time
homogeneous case. We will also see that the result from [22] enables us to
show (1.11) for observable functions that are allowed to have exponential
growth at infinity, while in the time homogeneous case [34] the estimate is
valid for observable functions growing at infinity not faster than a
polynomial. However, due to the interaction between the mixing of the solution
process and the irrational rotation inherited from the quasi-periodic force,
the convergence rate in our context cannot be arbitrarily close to the likely
optimal rate usually obtained in the time homogeneous case. Indeed, from the
proof, one can see that estimate (1.11) is a mixture of the convergence rate
to the variance of the time inhomogeneous solution process, and the
convergence rate of the Birkhoff sum for the irrational rotation with a
particular observable function involving the quasi-periodic invariant measure
that is related to the Diophantine approximation property of the frequency.
The combination of the two rates prohibits the possibility of
$\varepsilon_{0}$ being arbitrarily small. Besides, estimate (1.10) is
obtained by combining the martingale approximation and the Borel-Cantelli
lemma with an estimation on the convergence rate for the moments of the time
average of the observations centered by the quasi-periodic invariant measure.

###  1.2. Relation to existing works

(1) There is an extensive literature on the unique ergodicity and mixing of
invariant measures for time homogeneous infinite dimensional systems. However,
this literature is too vast a subject for us to review here. Hence we limit
ourselves to topics on the time inhomogenuous setting. The concept of quasi-
periodic invariant measure with its ergodicity and mixing for stochastic 2D
Navier-Stokes equation, is introduced in the present work. However, the quasi-
periodic invariant measure for ordinary stochastic differential equations has
been introduced in [17]. In fact, the probability measure valued path
$\mu_{t}$ satisfying (1.2) has been studied by Dynkin in [14, 15] where it is
called an entrance law. In this context, a quasi-periodic invariant measure is
an entrance law that possesses additional dynamical structure inherited from
the quasi-periodic force. The ergodicity and mixing for periodic invariant
measures, were first investigated and proved in [10] for 2D stochastic Navier-
Stokes equations with nondegenerate noise. In the hypoelliptic (extremely
degenerate noise) case, the ergodicity and mixing of the periodic invariant
measure were proved in the author’s recent work [30] under the condition of
the Grashof number being small. Our first result here is a sequel of [30] with
a deterministic force having multiple frequencies and does not place any
condition on the viscosity $\nu>0$ and the deterministic force other than some
spatial regularity.

(2) The martingale limit theory has been widely studied as a generalization of
the limit theory for the sum of independent random variables, see the
monograph [22]. It becomes a powerful tool when studying the limit theorems of
Markov processes, in the case that the Gordin’s martingale approximation is
available. This approach has been summarized in [28, 34] in the case of
uniformly mixing time homogeneous Markov processes with applications to
randomly forced PDE’s including the 2D Navier-Stokes equation. As we mentioned
above, the result in [28, 34] cannot be applied directly here due to the time
in-homogeneity and non-mixing feature of the homogenized process.

(3) For discrete time inhomogeneous Markov chains, progress on limit theorems
has been made over the decades, which is best summarized in [13]. However, the
problem and method we consider here, are different from that in [13] where
they mainly focus on the local limit theorems in the stationary regime for
discrete time Markov chains and take a different perspective. Furthermore,
despite the uniform boundedness condition on the observables, the uniform
elliptic condition in [13] does not hold in our hypoelliptic setting so that
the results cannot be applied. In fact, we are not aware of any existing
results on limit theorems for continuous time inhomogeneous Markov processes,
as well as applications to randomly forced PDE’s both in elliptic and
hypoelliptic settings.

##  2\. Settings and main results

###  2.1. Well-posedness, quasi-periodic invariant measures and quasi-periodic
solutions

Let $(M,d)$ be a metric space with metric $d$ and $C_{b}(\mathbb{R},M)$ the
space of bounded continuous functions endowed with the uniform convergence
topology generated by the following metric

$\underline{d}(q_{1},q_{2})=\sup_{t\in\mathbb{R}}d(q_{1}(t),q_{2}(t)).$

######  Definition 2.1 (Quasi-periodic functions).

A function $q\in C_{b}(\mathbb{R},M)$ is quasi-periodic with frequency
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\in\mathbb{R}^{n}$ if there
is $Q\in C(\mathbb{T}^{n},M)$ such that

(2.1) $\displaystyle q(t)=Q(\alpha
t)=Q(\alpha_{1}t,\alpha_{2}t,\cdots,\alpha_{n}t),$

where $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}$ are rationally independent
real numbers and $\mathbb{T}^{n}=\mathbb{R}^{n}/(2\pi)\mathbb{Z}^{n}$ is the
$n$-dimensional torus.

We also need the notion of a Diophantine condition on the frequency $\alpha$
as in [26].

######  Definition 2.2 (Diophantine condition).

A frequency $\alpha\in\mathbb{T}^{n}$ is said to satisfy a Diophantine
condition if there exist $K>0$ and $A>n$ such that

(2.2)
$\displaystyle\mathrm{dist}(k\cdot\alpha,\mathbb{Z})\geq\frac{K}{\|k\|^{A}},$

for all $k\in\mathbb{Z}^{n}$ with $\|k\|\neq 0$, where
$\displaystyle\|k\|:=\max_{1\leq i\leq n}|k_{i}|$, and
$k\cdot\alpha=k_{i}\alpha_{1}+k_{2}\alpha_{2}+\cdots+k_{n}\alpha_{n}$.

Consider the Navier-Stokes equation on the two dimensional torus
$\mathbb{T}^{2}$ driving by a deterministic quasi-periodic force and a
degenerate stochastic force. In the vorticity form, it reads

(2.3) $\displaystyle
dw(t,x)+B(\mathcal{K}w,w)(t,x)dt=\nu\mathrm{\mathrm{\Delta}}w(t,x)dt+f(t,x)dt+GdW(t),\quad
t>s,\quad w(s)=w_{0},$

where $s\in\mathbb{R}$, $w(t,x)\in\mathbb{R}$ denotes the value of the scalar
vorticity field, $\mathrm{\mathrm{\Delta}}$ is the Laplacian operator with
periodic boundary conditions, $B(\mathcal{K}w,w)=(\mathcal{K}w)\cdot\nabla w$
is the nonlinear term and $\mathcal{K}w$ is the velocity that is recovered
from $w$ by the Biot-Savart integral operator $\mathcal{K}$. The deterministic
initial condition $w_{0}$ lies in the state space of system (2.3), which is
chosen as
$H:=\left\\{w\in\mathrm{L}^{2}\left(\mathbb{T}^{2},\mathbb{R}\right):\int_{\mathbb{T}^{2}}wdx=0\right\\}$,
where the norm is denoted by $\left\|\cdot\right\|$ and the inner product is
$\langle\cdot,\cdot\rangle$. We also define the interpolation spaces
$H_{s}=\left\\{w\in
H^{s}\left(\mathbb{T}^{2},\mathbb{R}\right):\int_{\mathbb{T}^{2}}wdx=0\right\\}$
and the corresponding norms $\left\|\cdot\right\|_{s}$ by
$\|w\|_{s}=\left\|\left(-\mathrm{\Delta}\right)^{s/2}w\right\|$. Here $W(t)$
is a standard $d$ dimensional two-sided Brownian motion 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
$\mathcal{F}_{t}$ the filtration of $\sigma$-algebras generated by $W(t)$. The
coefficient of the noise is a bounded linear operator
$G:\mathbb{R}^{d}\rightarrow H_{\infty}:=\bigcap_{s>0}H_{s}$, such that
$Ge_{i}=g_{i}$ , where $\\{e_{i}\\}$ is the standard basis of $\mathbb{R}^{d}$
and $g_{i}\in H_{\infty}$ for $i=1,2,\cdots,d$. Then the noise can be
expressed as $GW(t)=\sum_{i=1}^{d}g_{i}W_{i}(t)$. Let
$\mathcal{B}_{0}:=\sum_{i=1}^{d}\|g_{i}\|^{2}$ be the energy input from the
noise.

We assume that the deterministic force $f\in
C_{b}\left(\mathbb{R},H_{2}\right)$ is quasi-periodic with frequency
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$. Then there is a function
$\Psi\in C(\mathbb{T}^{n};H_{2})$ such that $f(t,x)=\Psi(\alpha t,x)$. The
regularity condition imposed on $f$ is to ensure the spatial regularity of the
solution that is needed to show the asymptotic smoothing effect of the
dynamics in subsection 3.1.

The existence and uniqueness of the solution to equation (2.3) is well known,
see for example [3] and references therein. To be specific, we have the
following

######  Proposition 2.3 (well-posedness [3]).

Assume $f\in C_{b}(R,H_{2})$ and $g_{i}\in H_{\infty}$, $1\leq i\leq d$. For
any initial time $s\in\mathbb{R}$, every $T>s$ and $w_{0}\in H$, equation
(2.3) under the above conditions has a unique strong solution
$w(t,\omega;s,w_{0})$, adapted to the filtration $\mathcal{F}_{t}$, which
generates a stochastic flow $\mathrm{\Phi}(t,\omega;s,\cdot):H\rightarrow H$
such that $\mathrm{\Phi}(t,\omega;s,w_{0})=w(t,\omega;s,w_{0})$ for $s\leq
t,w_{0}\in H$ and

$w\in C\left([s,T];H\right)\cap
C\left((s,T];H_{3}\right),\,\,\mathbf{P}-\mathrm{a.s}..$

Here by a stochastic flow $\mathrm{\Phi}(t,\omega;s,w_{0})$, we mean that it
is a modification of the solution of equation (2.3) satisfying the following
conditions:

  1. (1)

It is adapted to $\mathcal{F}_{t}$ and for almost all $\omega$,
$\mathrm{\Phi}(t,\omega;s,w_{0})$ is continuous in $(t,s,w_{0})$ and
$\mathrm{\Phi}(s,\omega;s,w_{0})=w_{0}.$

  2. (2)

For almost all $\omega$,

$\mathrm{\Phi}(t+\tau,\omega;s,w_{0})=\mathrm{\Phi}(t+\tau,\omega;t,\mathrm{\Phi}(t,\omega;s,w_{0})),$

for $s\leq t$, $\tau>0$ and $w_{0}\in H$.

Throughout this work, we will also use $w_{s,t}(\omega,w_{0})$ or
$\Phi_{s,t}(\omega,w_{0})$ to represent the solution $w(t,\omega;s,w_{0})$.

The solution to (2.3) generates a two parameter Markov transition operator
$\mathcal{P}_{s,t}:B_{b}(H)\rightarrow B_{b}(H)$ defined by

(2.4)
$\displaystyle\mathcal{P}_{s,t}\varphi(w_{0}):=\mathbf{E}[\varphi(w(t;s,w_{0}))],\quad\varphi\in
B_{b}(H),$

where $B_{b}(H)$ is the space of bounded Borel measurable functions on $H$
with the supremum norm. We denote the transition probabilities as
$\mathcal{P}_{s,t}(w,A):=\mathcal{P}_{s,t}\mathbb{I}_{A}(w)$ for
$A\in\mathfrak{B}$, the Borel $\sigma$-algebra of $H$, where $\mathbb{I}_{A}$
is the characteristic function of $A$. By duality, the transition operator
$\mathcal{P}_{s,t}^{*}$ acts on the space $\mathcal{P}(H)$ of probability
measures on $H$ by

$\mathcal{P}_{s,t}^{*}\mu(A)=\int_{H}\mathcal{P}_{s,t}(w,A)\mu(dw),\text{ for
}\mu\in\mathcal{P}(H),A\in\mathfrak{B}.$

For $\eta>0$ small, recall the metric $\rho$ on $H$ weighted by the Lyapunov
function $e^{\eta\|w\|^{2}}$ as in (1.1),

(2.5)
$\displaystyle\rho(w_{1},w_{2})=\inf_{\gamma}\int_{0}^{1}e^{\eta\|\gamma(t)\|^{2}}\|\dot{\gamma}(t)\|dt,\quad\forall
w_{1},w_{2}\in H,$

where the infimum is taken over all differentiable path $\gamma$ connecting
$w_{1}$ and $w_{2}$. This metric naturally induces a Wasserstein metric
(allowed to take values in $[0,\infty]$) on $\mathcal{P}(H)$ by

(2.6)
$\displaystyle\rho(\mu_{1},\mu_{2})=\inf_{\mu\in\mathcal{C}(\mu_{1},\mu_{2})}\int_{H\times
H}\rho(u,v)\mu(dudv),$

where $\mathcal{C}(\mu_{1},\mu_{2})$ is the set of couplings of
$\mu_{1},\mu_{2}\in\mathcal{P}(H)$. The subset

(2.7)
$\displaystyle\mathcal{P}_{1}(H):=\\{\mu\in\mathcal{P}(H):\rho(\mu,\delta_{0})<\infty\\}$

is complete under the metric $\rho$ [6, 36], where $\delta_{0}$ is the Dirac
measure at $0$. For this Wasserstein metric, the following Monge-Kantorovich
duality is well-known [6, 36],

(2.8)
$\displaystyle\rho\left(\mu_{1},\mu_{2}\right)=\sup_{\mathrm{Lip}_{\rho}(\phi)\leq
1}\left|\int\phi(x)\mu_{1}(dx)-\int\phi(x)\mu_{2}(dx)\right|,\quad\forall\mu_{1},\mu_{2}\in\mathcal{P}_{1}(H),$

where $\mathrm{Lip}_{\rho}(\phi)$ is the Lipschitz constant of the function
$\phi$ on $H$ endowed with the metric $\rho$.

######  Definition 2.4 (Quasi-periodic invariant measures).

A quasi-periodic invariant measure of system (2.3) is a quasi-periodic
function $\mu\in C(\mathbb{R},\mathcal{P}(H))$ that is invariant under the
Markov transition operators

$\displaystyle\int_{H}\mathcal{P}_{s,t}\varphi(w)\mu_{s}(dw)=\int_{H}\varphi(w)\mu_{t}(dw),\quad
s\leq t,\quad\varphi\in B_{b}(H),$

or equivalently

$\displaystyle\mathcal{P}_{s,t}^{*}\mu_{s}=\mu_{t},\quad s\leq t.$

We also give the following definition of the random quasi-periodic solution in
the path wise sense.

######  Definition 2.5 (Random quasi-periodic solutions).

A random quasi-periodic solution of system (2.3) is a progressively measurable
stochastic process $w^{*}(t,\omega)$ defined on $\mathbb{R}\times\Omega$ that
satisfies the following property:

  1. (1)

(Invariance property) $\Phi(t,\omega;s,w^{*}(s,\omega))=w^{*}(t,\omega)$, for
almost all $\omega$;

  2. (2)

(Quasi-periodicity) $w^{*}(t,\theta_{-t}\omega)$ is a quasi-periodic function
for almost every sample $\omega$. That is, there is a function
$Q:\mathbb{T}^{n}\times\Omega\rightarrow H$, where $Q(h,\omega)$ is continuous
in $h\in\mathbb{T}^{n}$ for almost every $\omega\in\Omega$ and measurable in
$\omega$ for each fixed $h$, such that for every $t\in\mathbb{R}$,

$w^{*}(t,\theta_{-t}\omega)=Q(\alpha t,\omega),\quad\mathbf{P}-\mathrm{a.s}..$

Here $\theta:\mathbb{R}\times\mathrm{\Omega}\rightarrow\mathrm{\Omega}$ given
by

(2.9) $\displaystyle\theta_{t}\omega(\cdot):=\omega(t+\cdot)-\omega(t)$

is the Wiener shift.

###  2.2. Homogenization through skew product

To deal with the time inhomogeneity, we adopt a classical method that has been
widely used in the study of non-autonomous problems arising from deterministic
differential equations and dynamical systems. Let $\mathcal{H}(f)$ be the
closure in $C_{b}(\mathbb{R},H_{2})$ of $\\{f(\cdot+s)|s\in\mathbb{R}\\}$, the
set of time translations of $f$. The set $\mathcal{H}(f)$ is called the hull
of $f$, which is compact since $f$ is quasi-periodic. It is the minimal
invariant set of the translation group $\\{T(t)\\}$ acting on
$C_{b}(\mathbb{R};H_{2})$ by $(T(t)g)(s)=g(t+s)$, which is called the Bebutov
shift flow. The Bebutov shift dynamics has the advantage of capturing the
nonautonomy caused by the quasi-periodicity of $f$ and the compactness of the
hull $\mathcal{H}(f)$ allows us to apply tools from dynamical systems to
analyze related problems. For each $g\in H(f)$, there is a unique solution
$w_{s,t,g}(\omega,w_{0})$ of (2.3) by replacing $f$ with $g$, which is called
the process corresponding to problem (2.3) with time symbol $g$ [8]. The
homogenization process associated with $\Phi_{s,t}(\omega,w_{0})$ is then the
homogeneous Markov process in the extended phase space $H\times\mathcal{H}(f)$
defined by $S(t,\omega,w_{0},g):=\left(w_{0,t,h}(\omega,w_{0}),T(t)g\right)$.

In the quasi-periodic case, it turns out that $\mathcal{H}(f)=\\{\Psi(\alpha
t+h_{0},x)|h_{0}\in\mathbb{T}^{n}\\}$ [8], where $\Psi\in
C(\mathbb{T}^{n},H_{2})$ is the function corresponding to $f$. Hence it is
more convenient to work with $\mathbb{T}^{n}$ instead of $\mathcal{H}(f)$. The
irrational rotation flow $\beta_{t}h:=h+\alpha t$ on torus corresponds to the
Bebutov shift $T(t)$ on $\mathcal{H}(f)$ through the continuous map $\Psi$.
Instead of working on $H\times\mathcal{H}(f)$, we will study the associate
homogeneous Markov process $X_{s,t}(w_{0},h_{0})$ on $H\times\mathbb{T}^{n}$
given by the solution of the following equation

(2.10)
$\left\\{\begin{array}[]{l}dw(t,x)+B(\mathcal{K}w,w)(t,x)dt=\nu\mathrm{\mathrm{\Delta}}w(t,x)dt+\Psi(\beta_{t},x)dt+GdW(t),\\\
d\beta_{t}=\alpha dt,\\\ w(s)=w_{0},\beta_{s}=h_{0}.\end{array}\right.$

Note that the solution
$X_{s,t}(w_{0},h_{0})=(\Phi_{s,t,\beta_{-s}h_{0}}(w_{0}),\beta_{t-s}h_{0})$
for $(w_{0},h_{0})\in H\times\mathbb{T}^{n}$, where
$\Phi_{s,t,\beta_{-s}h_{0}}(w_{0})$ is the solution to $\eqref{NS}$ by
replacing $f(t,x)=\Psi(\alpha t,x)$ with $\Psi(\alpha t-\alpha s+h_{0},x)$. We
will use $w_{s,t,h}(w_{0})$ to denote the solution of (2.3) with $f(t,x)$
replaced by $\Psi(\beta_{t},x)$ with $\beta_{t}h=h+\alpha t$.

Let
$\mathcal{P}_{s,t,h}\phi(w_{0})=\mathbf{E}\phi\left(\Phi_{s,t,h}(w_{0})\right)$
be the transition operator corresponding to the time symbol $h$. In
particular, $\mathcal{P}_{s,t,0}=\mathcal{P}_{s,t}$ as defined in (2.4). It
follows from the uniqueness of solution that the following translation
identity holds:

(2.11)
$\displaystyle\mathcal{P}_{s+\tau,t+\tau,h}=\mathcal{P}_{s,t,\beta_{\tau}h},\,\tau\in\mathbb{R},\,h\in\mathbb{T}^{n}.$

For $\varphi\in B_{b}(H\times\mathbb{T}^{n})$, the Markov transition operator
associated with the homogenized process is given by

(2.12) $\displaystyle
P_{s,t}\varphi(w_{0},h_{0})=\mathbf{E}\varphi\left(X_{s,t}\right)=\mathbf{E}\varphi\left(\Phi_{s,t,\beta_{-s}h_{0}}(w_{0}),\beta_{t-s}h_{0}\right)=\mathcal{P}_{s,t,\beta_{-s}h_{0}}\varphi(\cdot,\beta_{t-s}h_{0})(w_{0}).$

In view of the translation identity (2.11) and time homogeneity of $X_{s,t}$,
we can assume the initial time of the homogenized process $s=0$ for
simplicity.

Notations. $\langle\mu,\phi\rangle$ represents the integration of a real
valued function $\phi$ on $H$ with respect to a given measure $\mu$. Denote
$\mathbf{E}[X,Y\geq Z]=\mathbf{E}X\mathbb{I}_{\\{Y\geq Z\\}}$ for given random
variables $X,Y,Z$. Also $\mathrm{Lip}_{\rho}(H)$ is the space of Lipschitz
functions on $H$ endowed with the metric $\rho$ as in (2.5) and
$\mathrm{Lip}_{\rho}(\phi)$ is the Lipschitz constant for
$\phi\in\mathrm{Lip}_{\rho}(H)$. The constant $C$ could be different from line
to line, but we will emphasize its dependence on the parameters when
necessary. $X_{t}$ represents the solution $X_{s,t}$ of (2.10) with $s=0$ and
$P_{t}$ means $P_{0,t}$. Given any functional $\varphi$ of the process
$X_{t}$, the expectation $\mathbf{E}_{(w,h)}\varphi(X_{t})$ means
$\mathbf{E}\varphi(X_{t}(w,h))$, i.e. the expectation of the random variable
$\varphi(X_{t}(w,h))$ with respect to the Wiener measure $\mathbf{P}$.

###  2.3. Main results

In this subsection, we formulate the main results of the present paper in
details. We use the metric (2.6) on $\mathcal{P}(H)$ to measure the
convergence to the quasi-periodic invariant measure. The reason for working
with the Wasserstein metric is that the transition probabilities in infinite
dimensional systems are likely to be mutually singular, especially when the
strong Feller property does not hold [24]. Hence the convergence to the
invariant measure often fails under the total variation metric and one would
like to replace it by a weaker Wasserstein metric. Note that the metric $\rho$
weighted by the Lyapunov function $V(w)=e^{\eta\|w\|^{2}}$ depends on the
parameter $\eta>0$.

The following Theorem 2.6-2.9 are our main results under the standing
assumption:

(2.13) $\displaystyle f\in C_{b}(R,H_{2})\text{ is quasi-periodic};g_{i}\in
H_{\infty},\forall 1\leq i\leq d;\text{ and }A_{\infty}=H.$

The first result is the following unique ergodicity and exponentially mixing
of the quasi-periodic invariant measure for (2.3) under the Wasserstein metric
(2.6).

######  Theorem 2.6 (Ergodicity and mixing).

There is a unique quasi-periodic invariant measure $\mu_{t}$ for (2.3) given
by a unique map $\Gamma\in C(\mathbb{T}^{n},\mathcal{P}(H))$, i.e.,
$\mu_{t}=\Gamma_{\beta_{t}0}$, and $\mathcal{P}_{s,t}^{*}\mu_{s}=\mu_{t}$.
Moreover, there exists $\eta_{0}>0$, such that for every
$\eta\in(0,\eta_{0}]$, there are constants $C,\varpi>0$, such that $\Gamma\in
C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$ and

(2.14) $\displaystyle\rho(\mathcal{P}_{s,s+t}^{*}\mu,\mu_{s+t})\leq
Ce^{-\varpi t}\rho(\mu,\mu_{s}),\quad\forall s\in\mathbb{R},t\geq
0,\mu\in\mathcal{P}(H),$

where $C,\varpi$ do not depend on $s$. Furthermore, $\Gamma\in
C^{\zeta}(\mathbb{T}^{n},(\mathcal{P}_{1}(H),\rho))$ if $\Psi\in
C^{\gamma}(\mathbb{T}^{n},H)$, where $\zeta=\frac{\varpi\gamma}{r+\varpi}$
with $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ from estimate
(6.3).

In an equivalent form that involves the transition operator acting on
observables, we have for every $\phi\in C_{\eta}^{1}$,

(2.15)
$\displaystyle\left\|\mathcal{P}_{s,s+t}\phi-\int_{H}\phi(w)\mu_{s+t}(dw)\right\|_{\eta}\leq
Ce^{-\varpi t}\|\phi\|_{\eta}.$

Here

$C_{\eta}^{1}:=\left\\{\phi\in C^{1}(H):\|\phi\|_{\eta}:=\sup_{w\in
H}e^{-\eta\|w\|^{2}}\big{(}|\phi(w)|+\|\nabla\phi(w)\|\big{)}<\infty\right\\}.$

Theorem 2.6 will be proved in Section 3, by combining a fixed point argument
with the following uniform contraction property proved in the same section.

######  Theorem 2.7.

$($Contraction on $\mathcal{P}(H)$$)$ There exists $\eta_{0}>0$ such that for
$\eta\in(0,\eta_{0}]$, there are positive constants $C$ and $\varpi$ such that

(2.16)
$\displaystyle\rho(\mathcal{P}_{s,s+t,h}^{*}\mu_{1},\mathcal{P}_{s,s+t,h}^{*}\mu_{2})\leq
Ce^{-\varpi t}\rho(\mu_{1},\mu_{2}),$

for every $s\in\mathbb{R}$, $t\geq 0$, $h\in\mathbb{T}^{n}$ and any
$\mu_{1},\mu_{2}\in\mathcal{P}(H)$.

The second result is on the strong law of large numbers (SLLN) and the central
limit theorem (CLT) for the solution process. The proof will be given in
Section 4. To state the results, we first define the space of observable
functions. For $\gamma\in(0,1]$, let $C^{\gamma}_{\eta}(H)$ be the space of
Hölder continuous functions with finite norms weighted by the Lyapunov
function $e^{\eta\|w\|^{2}}$,

(2.17) $\displaystyle
C^{\gamma}_{\eta}(H):=\left\\{\phi:H\rightarrow\mathbb{R}:\|\phi\|_{\gamma,\eta}<\infty\right\\},$

where

$\displaystyle\|\phi\|_{\gamma,\eta}:=\sup_{w\in
H}\frac{|\phi(w)|}{e^{\eta\|w\|^{2}}}+\sup_{0<\|w_{1}-w_{2}\|\leq
1}\frac{|\phi(w_{1})-\phi(w_{2})|}{\|w_{1}-w_{2}\|^{\gamma}\left(e^{\eta\|w_{1}\|^{2}}+e^{\eta\|w_{2}\|^{2}}\right)}.$

Recall that $w_{s,s+t}(w_{0})$ is the solution to (2.3) starting from
$w_{0}\in H$ at time $s\in\mathbb{R}$.

######  Theorem 2.8.

There is a constant $\eta_{0}>0$, such that for every $\eta\in(0,\eta_{0}]$,
$\phi\in C^{\gamma}_{\eta}(H)$ and $w_{0}\in H$, we have

1\. SLLN for the time inhomogeneous solution process:

(2.18)
$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle\Big{)}dt\overset{\mathrm{a.s.}}{\longrightarrow}0.$

2\. CLT for the time inhomogeneous solution process:

(2.19)
$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{\sqrt{T}}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle\Big{)}dt\overset{\mathcal{D}}{=}N(0,\sigma^{2}),$

where $N(0,\sigma^{2})$ a centered normal variable with variance $\sigma^{2}$
and $\mathcal{D}$ represents the convergence in distribution. The variance

$\sigma^{2}=\sigma_{\phi}^{2}=\lim_{T\rightarrow\infty}\frac{1}{T}\mathbf{E}\left[\int_{0}^{T}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle
dt\right]^{2},$

where $\sigma_{\phi}\geq 0$ is independent of $s$.

The third result is an estimate of the rate of convergence in the limit
theorems. The proof will be given in Section 4.

######  Theorem 2.9.

There is a constant $\eta_{0}>0$ such that the following estimates hold.

1\. (Rate of convergence in SLLN) Let $\varepsilon>0$, for every integer
$p\geq 3$ satisfying $2^{p}>1/\varepsilon$, every
$\eta\in(0,2^{-p-1}\eta_{0}]$, and every $\phi\in C^{\gamma}_{\eta}(H)$,
$w_{0}\in H$, $s\in\mathbb{R}$, there is an almost surely finite random time
$T_{0}(\omega)\geq 1$, depending on
$p,\varepsilon,\|\phi\|_{\gamma,\eta,H},s,\|w_{0}\|$ such that for all
$T>T_{0}$, we have

$\displaystyle\left|\frac{1}{T}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\big{\langle}\mu_{s+t},\phi\big{\rangle}\Big{)}dt\right|\leq
CT^{-\frac{1}{2}+\varepsilon},$

where $C>0$ is a constant that does not depend on the above parameters.
Moreover, for every $0<\ell<\min\\{2^{p}\varepsilon-1,2^{p-2}-1\\}$, there is
a constant $C_{p}=C_{p}(\|\phi\|_{\gamma,\eta,H},\ell,\varepsilon)$ such that

$\mathbf{E}T_{0}^{\ell}\leq C_{p}e^{2^{p+1}\eta\|w_{0}\|^{2}}.$

2\. (Rate of convergence in CLT) Assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$
and the frequency $\alpha$ satisfies the Diophantine condition (2.2) with
constant $A$ and dimension $n$ (of the torus). Let $\Phi_{\sigma}$ be the
distribution function of $N(0,\sigma^{2})$. Also let
$\Lambda=\frac{\varpi}{5(2-\gamma)}$, $\zeta=\frac{\varpi\gamma}{r+\varpi}$
and $\overline{\gamma}_{0}=\frac{\Lambda\zeta}{5(\Lambda+r)(2-\gamma)}$, where
$\varpi$ is the mixing rate from Theorem 2.6 and
$r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ is the constant
from (6.3).

(1). For any integer $p\geq 2$, $\eta\in(0,2^{-p-1}\eta_{0}]$, and $\phi\in
C^{\gamma}_{\eta}(H)$ with $\sigma_{\phi}^{2}>0$, and $w_{0}\in H$, there are
constants $C_{p}=C_{p}(\|\phi\|_{\gamma,\eta},\|w_{0}\|)>0$ and $T_{0}>0$ such
that for all $T\geq T_{0}$,

$\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left\\{\frac{1}{\sqrt{T}}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle\Big{)}dt\leq
z\right\\}-\Phi_{\sigma_{\phi}}(z)\right|\leq
C_{p}\left(T^{-\frac{1}{4}}+T^{-\frac{2^{p-2}}{2^{p}+1}}+T^{-\frac{2^{p-1}\overline{\gamma}_{0}}{(2^{p}+1)(A+n)}}\right),$

(2). For $\eta\in(0,2^{-7}\eta_{0}]$ and $\phi\in C^{\gamma}_{\eta}(H)$ such
that $\sigma_{\phi}^{2}=0$, and $w_{0}\in H$, there is a constant
$C=C(\|\phi\|_{\gamma,\eta},\|w_{0}\|)>0$ such that for all $T\geq 1$,

$\displaystyle\sup_{z\in\mathbb{R}}\left(|z|\wedge
1\right)\left|\mathbf{P}\left\\{\frac{1}{\sqrt{T}}\int_{0}^{T}\Big{(}\phi(w_{s,s+t}(w_{0}))-\langle\mu_{s+t},\phi\rangle\Big{)}dt\leq
z\right\\}-\Phi_{0}(z)\right|\leq
C\left(T^{-\frac{1}{4}}+T^{-\frac{\overline{\gamma}_{0}}{2(A+n)}}\right),$

The following result shows that if the viscosity is large, then the dynamics
of system (2.3) is actually trivial. Let $c_{0}$ be the constant from the
Ladyzhenskaya’s inequality

(2.20) $\displaystyle||w||_{\mathrm{L}^{4}(\mathbb{T}^{2})}^{2}\leq
c_{0}\|w\|_{1}\|w\|.$

And let $G=\sqrt{\|f\|_{\infty}^{2}/\nu^{4}+\mathcal{B}_{0}/\nu^{3}}$ be the
Grashof number for the whole system, where
$\|f\|_{\infty}=\sup_{t\in\mathbb{R}}\|f(t)\|$. The following theorem will be
proved in Section 5. Note that we do not need the assumption $A_{\infty}=H$
here.

######  Theorem 2.10.

Let $\overline{\eta}>0$. Assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$. If

(2.21) $\displaystyle Gc_{0}\leq\sqrt{1/2},\,\text{ and
}\,\nu^{3}>8(n+\overline{\eta})c_{0}^{2}\mathcal{B}_{0}\gamma^{-1},$

then there exists a random quasi-periodic solution $w^{*}(t,\omega)$ of
equation (2.3) in the sense of Definition 2.5, where the function
$Q(h,\omega)$ associated with $w^{*}(t,\theta_{-t}\omega)$ has a continuous
(with respect to $h$) modification that is $\eta$-Hölder continuous for all
$0<\eta<\frac{\overline{\eta}\gamma}{2(n+\overline{\eta})}$. Moreover,
$w^{*}(t,\omega)$ exponentially attracts all other solutions both in forward
and pullback times. The law of $w^{*}(t,\omega)$ gives the unique quasi-
periodic invariant measure.

######  Remark 2.1.

The quasi-periodic solution is obtained as the composition of a random field
$Q:\mathbb{T}^{n}\times\Omega$ with the irrational rotation flow on the torus
$\mathbb{T}^{n}$ and the Wiener shift on $\Omega$, i.e.
$w^{*}(t,\omega)=Q(\beta_{t}0,\theta_{t}\omega)$. In view of the quasi-
periodicity in the Definition 2.5, one requires the random field $Q$ to be
almost surely continuous in $\mathbb{T}^{n}$ component. The conditions imposed
in Theorem 2.10 are used to find the desired continuous modification of the
random field by applying the Kolmogorov continuity theorem. This is an
essential feature in the multiple frequency setting that is different from the
periodic case as in [30].

##  3\. Ergodicity and exponentially mixing

In this section, we will prove Theorem 2.6 and Theorem 2.7.

###  3.1. Contraction on $\mathcal{P}(H)$

We first prove Theorem 2.7 in this subsection, namely, the “uniform fiber-
wise” contraction property (2.16) of the transition operators
$\\{\mathcal{P}_{s,t,h}^{*}\\}_{h\in\mathbb{T}^{n}}$ when acting on
$\mathcal{P}(H)$. The result can be regarded as an extension of the Harris-
like theorem for infinite dimensional systems first established by Hairer and
Mattingly [24] in the time homogeneous setting. In [30], the result was
extended to the time inhomogeneous case when the deterministic force is time
periodic. In fact, the proof of Theorem 2.7 is a rework of the proof from [30]
in a fiber-wise manner, as long as we prove the following three “uniform
fiber-wise” estimates on the dynamics.

The first estimate is a Lyapunov structure that reveals the dissipation
property of system (2.3) and gives the contraction of (2.16) at large scales.

######  Proposition 3.1 (The Lyapunov Structure).

Let $V(w)=\exp(\eta\|w\|^{2})$, $\eta_{0}>0$ be the constant from (6.1) and
$\alpha(t)=e^{-\nu t}$. For any $\kappa\geq 1$, and any
$\eta\in(0,\frac{\eta_{0}}{2\kappa}]$, there is a constant $C>0$ such that for
any $s\in\mathbb{R}$, and $(w,h)\in H\times\mathbb{T}^{n}$, we have

(3.1) $\displaystyle\mathbf{E}V^{\kappa}(\Phi_{s,s+t,h}(w))\leq
CV^{\kappa\alpha(t)}(w),$ (3.2)
$\displaystyle\mathbf{E}\rho(0,w_{s,s+t,h}(w))^{\kappa}\leq CV^{2\kappa}(w),$

for all $t\geq 0$ and

(3.3)
$\displaystyle\mathbf{E}V^{\kappa}(\Phi_{s,s+t,h}(w))(1+\|\nabla\Phi_{s,s+t,h}(w))\xi\|)\leq
CV^{\kappa\alpha(t)}(w)$

for every $t\in[0,1],h\in\mathbb{T}^{n}$ and $\xi\in H$ with $\|\xi\|=1$.

###### Proof.

Inequality (3.1) is a reformulation of (6.1). The estimate (3.2) follows from
(6.1). Indeed, by the definition of $\rho$,

$\displaystyle\mathbf{E}\rho(0,w_{s,s+t,h}(w))^{\kappa}$
$\displaystyle\leq\mathbf{E}\|w_{s,s+t,h}(w)\|^{\kappa}V^{\kappa}(w_{s,s+t,h}(w))$
$\displaystyle\leq C\mathbf{E}V^{2\kappa}(w_{s,s+t,h}(w))\leq
C\exp(2\kappa\eta\|w\|^{2}),$

for $\eta\leq\frac{\eta_{0}}{2\kappa}$, where $\eta_{0}$ is from (6.1).
Following the proof of (6.4) or as in [30], one can show that for any
$\eta>0$, there is a constant $C>0$ such that

$\displaystyle\|\nabla\Phi_{s,s+t,h}(w)\xi\|\leq\exp\left(\nu\eta\int_{s}^{s+t}\|\Phi_{s,r,h}(w)\|_{1}^{2}dr+Ct\right).$

The estimate (3.3) then follows from (6.1) and (6.2). ∎

The second is a gradient inequality that shows the asymptotic smoothing effect
of the dynamics by the highly degenerate noise and the spatial Laplacian,
which is applied, together with the Lyapunov structure, to show the
contraction of (2.16) at small scales.

######  Proposition 3.2 (The gradient inequality).

There exists constant a $\eta_{0}>0$ such that for every $\eta\in(0,\eta_{0}]$
and $a>0$ there exists constants $C=C(\eta,a)>0$ and $p\in(0,1)$ such that

(3.4) $\displaystyle\left\|\nabla\mathcal{P}_{s,s+t,h}\phi(w)\right\|\leq
C\exp(p\eta\|w\|^{2})\left(\sqrt{\left(\mathcal{P}_{s,s+t,h}|\phi|^{2}\right)(w)}+e^{-at}\sqrt{\left(\mathcal{P}_{s,s+t,h}\|\nabla\phi\|^{2}\right)(w)}\right)$

for every Frechet differentiable function $\phi$, every $w\in
H,s\in\mathbb{R},t\geq 0$ and $h\in\mathbb{T}^{n}$. Here $C(\eta,a)$ does not
depend on $s,w,h$ and $\phi$.

###### Proof.

Note that we have assumed $A_{\infty}=H$ and $\Psi\in
C(\mathbb{T}^{n},H_{2})$. Hence for each $h\in\mathbb{T}^{n}$, inequality
(3.4) with a constant $C$ depending on
$\displaystyle\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h)\|$ follows from
Proposition 6.6 in [30] with the deterministic force $\Psi(\beta_{t}h)$. It
employs an infinite dimensional Malliavin calculus and an infinite dimensional
Hörmander type condition as in [23, 25] to investigate the propagation of the
degenerate noise throughout the phase space $H$, combined with a Foias-Prodi’s
type estimate, which yields the asymptotic smoothing estimate of the
transition operator as in (3.4). The constant $C$ is actually independent of
$h$ since
$\displaystyle\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h)\|=\sup_{g\in\mathbb{T}^{n}}\|\Psi(g)\|$.
∎

The last estimate is a particular form of irreducibility (slightly stronger
than the usual topological irreducibility) that concerns the small positive
probability of being contraction of the dynamics on the phase space. It deals
with the contraction (2.16) at intermediate scales.

######  Proposition 3.3 (Irreducibility).

For any $R,\varepsilon,T>0,$ and $r\in(0,1]$, there exists $a>0$ so that

(3.5) $\displaystyle\inf_{\|w_{1}\|,\|w_{2}\|\leq
R}\sup_{\Gamma\in\mathcal{C}\left(\mathcal{P}_{s,s+T,h}^{*}\delta_{w_{1}},\mathcal{P}_{s,s+T,h}^{*}\delta_{w_{2}}\right)}\Gamma\left\\{(w_{1}^{\prime},w_{2}^{\prime})\in
H\times H:\rho_{r}(w_{1}^{\prime},w_{2}^{\prime})<\varepsilon\right\\}\geq a,$

for all $s\in\mathbb{R},h\in\mathbb{T}^{n}$. Here the metric $\rho_{r}$ is
given by

$\displaystyle\rho_{r}(w_{1},w_{2})=\inf_{\gamma}\int_{0}^{1}V^{r}(\gamma(t))\|\dot{\gamma}(t)\|dt,\quad\forall
w_{1},w_{2}\in H,$

where the infimum is taken over all differentiable path $\gamma$ in $H$
connecting $w_{1}$ and $w_{2}$.

For the reader’s convenience, we give the proof of this proposition after the
proof of Theorem 2.7.

###### Proof of Theorem 2.7.

For each $h\in\mathbb{T}^{n}$, it follows from Lemma 3.15-3.17 in [30] that
Proposition 3.1-3.3 imply the corresponding contractions at different scales,
which in turn implies the contration (2.16) by following Theorem 2.2 in [30].
The constant $C$ is independent of $h\in\mathbb{T}^{n}$ since the estimates in
Proposition 3.1-3.3 are independent of $h$. The proof is complete. ∎

####  3.1.1. Irreducibility: The Proof of Proposition 3.3

This subsection is devoted to the proof the irreducibility in Proposition 3.3.
It is based on the topological irreducibility (Lemma 3.4) which is a
consequence of the celebrated controllability results of Agrachev and Sarychev
[1, 2], and a parabolic regularizing property (Lemma 3.5) of the Navier-Stokes
equations. We first state and prove the two lemmas and then give the proof of
Proposition 3.3 at the end.

The first lemma states that uniformly for the initial positions in any given
compact set, the system has a positive probability to enter into any
neighborhood of a given position.

######  Lemma 3.4.

For any compact set $K\subset H$, any $\delta,T>0$, and $v\in H$, there is
$a_{0}:=a_{0}(K,\delta,T,v)>0$ such that

$\displaystyle\inf_{w\in
K,h\in\mathbb{T}^{n}}\mathcal{P}_{0,T,h}(w,B_{\delta}(v))\geq a_{0}.$

###### Proof.

By the well known controllability argument of Agrachev and Sarychev [1, 2]
(see also [19]) and the condition $A_{\infty}=H$, it follows that for each
$h\in\mathbb{T}^{n}$ the Navier-Stokes equation (2.3) with the deterministic
force $f$ replaced by $\Psi(\beta_{t}h)$ is approximately controllable in $H$.
Hence it is topologically irreducible, i.e., for all $w,v\in H$,
$\delta>0,T>0$ there is $\varepsilon=\varepsilon(w,v,\delta,T,h)>0$ such that

$\displaystyle\mathcal{P}_{0,T,h}\mathbb{I}_{B_{\delta/2}(v)}(w)=\mathcal{P}_{0,T,h}(w,B_{\delta/2}(v))\geq\varepsilon.$

Now let $0\leq L_{\delta}\leq 1$ be a Lipschitz continuous function on $H$
such that $L_{\delta}(w)=1$ on $B_{\delta/2}(v)$ and $L_{\delta}(w)=0$ outside
of $B_{\delta}(v)$. Then

$\displaystyle\mathcal{P}_{0,T,h}\mathbb{I}_{B_{\delta}(v)}(w)\geq\mathcal{P}_{0,T,h}L_{\delta}(w)\geq\mathcal{P}_{0,T,h}\mathbb{I}_{B_{\delta/2}(v)}(w)\geq\varepsilon.$

The Lipschitz continuity of $L_{\delta}$ and estimates (6.3) and (6.4) implies
that the function

$(w,h)\rightarrow\mathcal{P}_{0,T,h}L_{\delta}(w)$

is continuous. Hence on the compact set $K\times\mathbb{T}^{n}$, it attains
the infimum at some $(w_{0},h_{0})$, which implies

$\inf_{(w,h)\in
K\times\mathbb{T}^{n}}\mathcal{P}_{0,T,h}\mathbb{I}_{B_{\delta}(v)}(w)\geq\inf_{(w,h)\in
K\times\mathbb{T}^{n}}\mathcal{P}_{0,T,h}L_{\delta}(w)=\mathcal{P}_{0,T,h_{0}}L_{\delta}(w_{0})\geq\varepsilon(w_{0},v,\delta,T,h_{0}).$

The proof is then completed by taking
$a_{0}=\varepsilon(w_{0},v,\delta,T,h_{0})$. ∎

The second lemma is a result of the parabolic regularizing effect, which
states that any given bounded set has a positive probability to be mapped into
a compact set by the solution map.

######  Lemma 3.5.

For any $R>0,T>0$, there are $R_{1}:=R_{1}(R,T)>0$ and
$a_{1}:=a_{1}(R,R_{1},T)>0$ such that

$\displaystyle\inf_{w\in
B_{R}(0),h\in\mathbb{T}^{n}}\mathcal{P}_{0,T,h}\left(w,\overline{B}_{R_{1}}^{H_{1}}(0)\right)\geq
a_{1},$

where $B_{R}(0)$ is the ball in $H$ centered at $0$ with radius $R$, and
$\overline{B}_{R_{1}}^{H_{1}}(0)$ is the closed ball in $H_{1}$ centered at
$0$ with radius $R_{1}$.

###### Proof.

Let $w_{0,t,h}(w_{0})$ be the solution of (2.3) with the deterministic force
$f$ replaced by $\Psi(\beta_{t}h)$. We eliminate the additive noise by
subtracting from (2.3) the following auxiliary Ornstein-Uhlenbeck process

$dV_{t}=\nu\mathrm{\Delta}V_{t}dt+GdW_{t},\quad V(0)=0.$

It is readily seen that the process $u_{t,h}:=w_{0,t,h}(w_{0})-V_{t}$ solves
the equation

(3.6)
$\displaystyle\partial_{t}u=\nu\mathrm{\Delta}u-B\left(\mathcal{K}\left(u+V_{t}\right),u+V_{t}\right)+\Psi(\beta_{t}h),\quad
u_{0,h}=w_{0}.$

For $\delta>0,T>0$, consider the event that the $d$-dimensional Wiener process
has small amplitude on a finite interval

(3.7)
$\displaystyle\mathrm{\Omega}_{\delta,T}=\left\\{\omega\in\mathrm{\Omega}:|W(\omega,t)|\leq\delta,\quad\text{for
all}\quad t\in[0,T]\right\\},$

where $|\cdot|$ denote the usual norm in $\mathbb{R}^{d}$. Then there exists
$a_{1}=a_{1}(\delta,T)$ such that

(3.8) $\mathbf{P}(\mathrm{\Omega}_{\delta,T})\geq a_{1}>0.$

By the properties of stochastic convolutions, one can show that the Ornstein-
Uhlenbeck process $V_{t}$ has the following property on the event
$\mathrm{\Omega}_{\delta,T}$.

######  Lemma 3.6.

[16] For any $\delta>0,T>0$, there exists a positive deterministic constant
$\varepsilon_{\delta,T}$ such that $\varepsilon_{\delta,T}\rightarrow 0$ as
$\delta\rightarrow 0$ for $T$ fixed, and

(3.9)
$\sup_{t\in[0,T]}\left\|V(t,\omega)\right\|_{1}\leq\varepsilon_{\delta,T},\text{
for all}\quad\omega\in\mathrm{\Omega}_{\delta,T}.$

Taking $H$ inner product with $2u$ on both sides of (3.6) and applying
standard estimates on the nonlinear term (see for example [23]), we have (we
write $u$ or $u_{t}$ for $u_{t,h}$ for notational simplicity and denote
$\|\Psi\|_{\infty}=\sup_{h\in\mathbb{T}^{n}}\|\Psi(h)\|$)

$\displaystyle\partial_{t}\|u\|^{2}$ $\displaystyle=\langle\nu\Delta
u,2u\rangle-\langle
B\left(\mathcal{K}\left(u+V_{t}\right),u+V_{t}\right),2u\rangle+\langle\Psi(\beta_{t}h),2u\rangle$
$\displaystyle\leq-2\nu\|u\|_{1}^{2}-2\langle
B\left(\mathcal{K}V_{t},V_{t}\right),u\rangle-2\langle
B\left(\mathcal{K}u,V_{t}\right),u\rangle+\frac{4}{\nu}\|\Psi\|_{\infty}^{2}+\nu\|u\|^{2}$
(3.10)
$\displaystyle\leq-\nu\|u\|_{1}^{2}+C\|V_{t}\|_{1}^{2}\|u\|+C\|u\|^{2}\|V_{t}\|_{1}+\frac{4}{\nu}\|\Psi\|_{\infty}^{2}$
$\displaystyle\leq
C\|V_{t}\|_{1}^{2}\|u_{t}\|^{2}+C(\|V_{t}\|_{1}^{4}+\|\Psi\|_{\infty}^{2}).$

Hence the Gronwall’s inequality and Lemma 3.6 yields

$\displaystyle\|u_{t}\|^{2}$
$\displaystyle\leq\|w_{0}\|^{2}\exp\left(C\int_{0}^{t}\|V_{\tau}\|_{1}^{2}d\tau\right)+C\int_{0}^{t}\left(\|V_{\tau}\|_{1}^{4}+\|\Psi\|_{\infty}^{2}\right)\exp\left(C\int_{\tau}^{t}\|V_{r}\|_{1}^{2}dr\right)d\tau$
(3.11) $\displaystyle\leq C(\|w_{0}\|^{2}+1)$

for all $t\in[0,T]$ and $\omega\in\Omega_{\delta,T}$, where the last constant
$C=C(\varepsilon_{\delta,T},T,\|\Psi\|_{\infty},\nu)$. Using this estimate and
integrating (3.10), we also have on $\Omega_{\delta,T}$,

(3.12)
$\displaystyle\nu\int_{0}^{T}\|u_{t}\|_{1}^{2}dt\leq\|w_{0}\|^{2}+\int_{0}^{T}\left(C\|V_{t}\|_{1}^{2}\|u_{t}\|+C\|u_{t}\|^{2}\|V_{t}\|_{1}+\frac{4}{\nu}\|\Psi\|_{\infty}^{2}\right)dt\leq
C(\|w_{0}\|^{2}+1)$

with constant $C$ only depending on
$\varepsilon_{\delta,T},T,\|\Psi\|_{\infty},\nu$.

Differentiating $t\|u_{t}\|_{1}^{2}$ with respect to $t$ and using (3.6)
yields

$\displaystyle\partial_{t}(t\|u_{t}\|_{1}^{2})$
$\displaystyle=\|u\|_{1}^{2}+2t\langle\partial_{t}u,-\Delta u\rangle$
$\displaystyle=\|u\|_{1}^{2}-2t\nu\|u\|_{2}^{2}+2t\left\langle
B\left(\mathcal{K}u,u\right)+B\left(\mathcal{K}V_{t},V_{t}\right)+B\left(\mathcal{K}u,V_{t}\right)+B\left(\mathcal{K}V_{t},u\right)-\Psi(\beta_{t}h),\Delta
u\right\rangle.$

It follows from standard estimates on the nonlinear term and interpolation
inequalities that

$\displaystyle|\langle B\left(\mathcal{K}u,u\right),\Delta u\rangle|$
$\displaystyle\leq C\|u\|\|u\|_{\frac{3}{2}}\|u\|_{2}\leq
C\|u\|\|u\|^{\frac{1}{4}}\|u\|_{2}^{\frac{3}{4}}\|u\|_{2}$ $\displaystyle\leq
C(\nu)\|u\|^{10}+\frac{\nu}{5}\|u\|_{2}^{2},$

and

$\displaystyle|\langle B\left(\mathcal{K}V_{t},V_{t}\right),\Delta
u\rangle|\leq C\|V_{t}\|_{1}^{2}\|u\|_{2}\leq
C(\nu)\|V_{t}\|_{1}^{4}+\frac{\nu}{5}\|u\|_{2}^{2},$ $\displaystyle|\langle
B\left(\mathcal{K}u,V_{t}\right),\Delta u\rangle|\leq
C\|V_{t}\|_{1}\|u\|_{1}\|u\|_{2}\leq
C(\nu)\|u\|_{1}^{2}\|V_{t}\|_{1}^{2}+\frac{\nu}{5}\|u\|_{2}^{2},$
$\displaystyle|\langle B\left(\mathcal{K}V_{t},u\right),\Delta u\rangle|\leq
C\|V_{t}\|_{1}\|u\|_{1}\|u\|_{2}\leq
C(\nu)\|u\|_{1}^{2}\|V_{t}\|_{1}^{2}+\frac{\nu}{5}\|u\|_{2}^{2},$
$\displaystyle|\langle\Psi(\beta_{t}h),\Delta
u\rangle|\leq\|\Psi\|_{\infty}\|u\|_{2}\leq
C(\nu)\|\Psi\|_{\infty}^{2}+\frac{\nu}{5}\|u\|_{2}^{2}.$

Therefore

$\displaystyle\partial_{t}(t\|u_{t}\|_{1}^{2})\leq\|u\|_{1}^{2}+C(\nu)t\left(\|u\|^{10}+\|V_{t}\|_{1}^{4}+\|u\|_{1}^{2}\|V_{t}\|_{1}^{2}+\|\Psi\|_{\infty}^{2}\right)$

Integrating this inequality from $0$ to $T$, using Lemma 3.6 and estimates
(3.11)-(3.12), one has

$\displaystyle\|u_{T,h}\|_{1}^{2}\leq C(1+\|w_{0}\|^{2})^{5},$

on $\Omega_{\delta,T}$, where $C$ depends on
$\varepsilon_{\delta,T},T,\|\Psi\|_{\infty},\nu$, but is independent of
$h\in\mathbb{T}^{n}$. Hence on $\Omega_{\delta,T}$ we have

$\|w_{0,T,h}(w_{0})\|_{1}^{2}\leq 2(\|u_{T,h}\|_{1}^{2}+\|V_{T}\|_{1}^{2})\leq
C(1+\|w_{0}\|^{10}),$

which completes the proof by taking $R_{1}=\sqrt{C(1+R^{10})}$. ∎

We are now in a position to give a proof of Proposition 3.3.

###### Proof of Proposition 3.3.

Fix any $v\in H$. For any $\delta,T>0$, $h\in\mathbb{T}^{n}$, and $w\in
B_{R}(0)$, we have by the Chapman-Kolmogorov relation, the translation
identity (2.11) that

$\displaystyle\mathcal{P}_{0,T,h}(w,B_{\delta}(v))=\int_{H}\mathcal{P}_{\frac{T}{2},T,h}(y,B_{\delta}(v))\mathcal{P}_{0,\frac{T}{2},h}(w,dy)=\int_{H}\mathcal{P}_{0,\frac{T}{2},\beta_{\frac{T}{2}}h}(y,B_{\delta}(v))\mathcal{P}_{0,\frac{T}{2},h}(w,dy).$

It follows from Lemma 3.5 that there are $R_{1}:=R_{1}(R,T/2)>0$, and
$a_{1}:=a_{1}(R,R_{1},T/2)>0$ such that

$\displaystyle\inf_{w\in
B_{R}(0),h\in\mathbb{T}^{n}}\mathcal{P}_{0,T/2,h}\left(w,\overline{B}_{R_{1}}^{H_{1}}(0)\right)\geq
a_{1},$

Combining this with Lemma 3.4 with $K=\overline{B}_{R_{1}}^{H_{1}}(0)$, we
have the existence of $a_{0}:=a_{0}(K,\delta,T,v)>0$ such that

$\displaystyle\inf_{w\in
B_{R}(0),h\in\mathbb{T}^{n}}\mathcal{P}_{0,T,h}(w,B_{\delta}(v))$
$\displaystyle\geq\inf_{w\in
B_{R}(0),h\in\mathbb{T}^{n}}\int_{K}\mathcal{P}_{0,\frac{T}{2},\beta_{\frac{T}{2}}h}(y,B_{\delta}(v))\mathcal{P}_{0,\frac{T}{2},h}(w,dy)$
$\displaystyle\geq a_{1}\inf_{y\in
K,h\in\mathbb{T}^{n}}\mathcal{P}_{0,\frac{T}{2},h}(y,B_{\delta}(v))\geq
a_{0}a_{1}.$

Therefore by the translation identity (2.11), one has

$\mathcal{P}_{s,s+T,h}(w,B_{\delta}(v))=\mathcal{P}_{0,T,\beta_{s}h}(w,B_{\delta}(v))\geq
a_{0}a_{1},\quad\forall w\in B_{R}(0),s\in\mathbb{R},h\in\mathbb{T}^{n}.$

Now for any $\varepsilon>0$, we choose
$\delta=\frac{\varepsilon}{2V(\|v\|+\varepsilon)}$, here
$V(w)=e^{\eta\|w\|^{2}}$ for $w\in H$ is the Lyapunov function from
Proposition 3.1. Then $\widetilde{B}_{\delta}(v):=B_{\delta}(v)\times
B_{\delta}(v)\subset\\{(w_{1}^{\prime},w_{2}^{\prime})\in H\times
H|\rho_{r}(w_{1}^{\prime},w_{2}^{\prime})<\varepsilon\\}$. Hence

$\displaystyle\inf_{w_{1},w_{2}\in
B_{R}(0)}\sup_{\Gamma\in\mathcal{C}\left(\mathcal{P}_{s,s+T,h}^{*}\delta_{w_{1}},\mathcal{P}_{s,s+T,h}^{*}\delta_{w_{2}}\right)}\Gamma\left\\{(w_{1}^{\prime},w_{2}^{\prime})\in
H\times H:\rho_{r}(w_{1}^{\prime},w_{2}^{\prime})<\varepsilon\right\\}$
$\displaystyle\geq\inf_{w_{1},w_{2}\in
B_{R}(0)}\mathcal{P}_{s,s+T,h}^{*}\delta_{w_{1}}\otimes\mathcal{P}_{s,s+T,h}^{*}\delta_{w_{2}}(\widetilde{B}_{\delta}(v))$
$\displaystyle=\inf_{w_{1},w_{2}\in
B_{R}(0)}\mathcal{P}_{s,s+T,h}(w_{1},B_{\delta}(v))\mathcal{P}_{s,s+T,h}(w_{2},B_{\delta}(v))\geq(a_{0}a_{1})^{2}.$

The proof is complete with $a=(a_{0}a_{1})^{2}$. ∎

###  3.2. Ergodicity and mixing

In this subsection, we prove Theorem 2.6 by applying a fixed point argument
and the uniform contraction (2.16) proved in the previous subsection. The
scheme is the same as that in [30], the difference is that the symbol space
here is $\mathbb{T}^{n}$ with $n\geq 1$, but in [30] the symbol space is a
circle. The idea is to regard the two parameter family of Markov transition
operators $\mathcal{P}_{s,t}^{*}$ as a non-autonomous dynamical system over
the space of probability measures, whose associated pull back mapping is a
contraction semigroup acting on an appropriate family of closed subsets of the
space of “quasi-periodic graphs” $C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$. And
the fixed point of the semigroup gives the desired quasi-periodic invariant
measure. Different from [30], we further show that this fixed point has a
Hölder regularity if the function $\Psi$ that generates the quasi-periodic
force does, and prove the weak convergence of time averages of the transition
probabilities to the unique invariant measure for the homogenized process.
These results play important roles in the study of the limit theorems in the
next section.

Note that Theorem 2.6 follows from the following Theorem 3.7 and Theorem 3.10
with $\mu_{s}=\Gamma_{\beta_{s}0}$ by taking $h=\beta_{s}0$ in (3.13) and
(3.16), and applying the translation identity (2.11).

######  Theorem 3.7.

There is a unique map $\Gamma\in C(\mathbb{T}^{n},\mathcal{P}(H))$, such that
$\mathcal{P}_{0,t,h}^{*}\Gamma_{h}=\Gamma_{\beta_{t}h}$ for any
$h\in\mathbb{T}^{n}$. Furthermore, there is a constant $\eta_{0}>0$, such that
for every $\eta\in(0,\eta_{0}]$, there are constants $C,\varpi>0$, such that
$\Gamma\in C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$ and

(3.13) $\displaystyle\rho(\mathcal{P}_{0,t,h}^{*}\mu,\Gamma_{\beta_{t}h})\leq
Ce^{-\varpi t}\rho(\mu,\Gamma_{h}),\quad t\geq
0,\mu\in\mathcal{P}(H),h\in\mathbb{T}^{n},$

where $C,\varpi$ dose not depend on $h$. Also
$\int_{H}\exp\left(2\kappa\eta\|w\|^{2}\right)\Gamma_{h}(dw)\leq C$ for all
$h\in\mathbb{T}^{n}$, where $\kappa\geq 2$ is the constant from Proposition
3.1.
Moreover, $\Gamma\in C^{\zeta}(\mathbb{T}^{n},(\mathcal{P}_{1}(H),\rho))$ if
$\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$, where
$\zeta=\frac{\varpi\gamma}{r+\varpi}$ with
$r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ from estimate
(6.3).

###### Proof.

Recall that $\mathcal{P}_{1}(H)$ is defined by (2.7). By Theorem 2.7, for any
$t\geq 0$, $\mathcal{P}_{0,t,h}^{*}$ maps $\mathcal{P}_{1}(H)$ to itself.
Denote for convenience

$\varphi:\mathbb{R}_{+}\times\mathcal{P}_{1}(H)\times\mathbb{T}^{n}\rightarrow\mathcal{P}_{1}(H),\text{
by }\varphi(t,\mu,h)=\mathcal{P}_{0,t,h}^{*}\mu.$

It follows from the translation identity (2.11) that $\varphi$ has the cocycle
property over the base dynamical system $(\mathbb{T}^{n},\mathbb{R},\beta)$
since for all $\tau,t\geq 0$ and $h\in{H(f)}$, $\mu\in\mathcal{P}_{1}(H)$,

$\mathcal{P}_{0,t+\tau,h}^{*}\mu=\mathcal{P}_{t,t+\tau,h}^{*}\mathcal{P}_{0,t,h}^{*}\mu=\mathcal{P}_{0,\tau,\beta_{t}h}^{*}\mathcal{P}_{0,t,h}^{*}\mu.$

Hence the pull-back map $S^{t}$ induced from $\varphi$, is defined on the
space of quasi-periodic graphs $C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$, i.e.,

$S^{t}(\gamma)(h):=\varphi(t,\gamma(\beta_{-t}h),\beta_{-t}h)),\,\gamma\in
C(\mathbb{T}^{n},\mathcal{P}_{1}(H)),$

which satisfies the semigroup property
$S^{t_{1}}S^{t_{2}}\gamma(h)=S^{t_{1}+t_{2}}\gamma(h)$. We would like to apply
the fixed point theorem for $S^{t}$ on $C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$
endowed with the metric (which is complete since $(\mathcal{P}_{1}(H),\rho)$
is complete.)

$p(\gamma_{1},\gamma_{2}):=\max_{h\in\mathbb{T}^{n}}\rho(\gamma_{1}(h),\gamma_{2}(h)),\,\gamma_{1},\gamma_{2}\in
C(\mathbb{T}^{n},\mathcal{P}_{1}(H)).$

However, the continuity of $\varphi(t,\mu,h)$ with respect to $(\mu,h)$ is
unclear due to the Lyapunov structure of the solution of (2.3). Hence
$C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$ may not be invariant under the map
$S^{t}$. Indeed as in [30], one can show from the estimate (6.1) and (6.3)
that

(3.14)
$\displaystyle\rho(\mathcal{P}_{0,t,h_{1}}^{*}\mu,\mathcal{P}_{0,t,h_{2}}^{*}\mu)\leq
Ce^{rt}\int_{H}g(w)\mu(dw)\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{1})-\Psi(\beta_{t}h_{2})\|,$

where $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ is from
(6.3), $g(w)=V^{2\kappa}(w)=\exp\left({2\kappa\eta}\|w\|^{2}\right)$, and the
Lyapunov function $V(w)=e^{\eta\|w\|^{2}}$ along with $\kappa,\eta$ are from
Proposition 3.1. It is unclear if each $\mu\in\mathcal{P}_{1}(H)$ yields
$\int_{H}g(w)\mu(dw)$ finite, therefore we confine ourselves to those measures
that make the integral finite to ensure the continuity. To be specific,
consider the family of closed subsets of $\mathcal{P}_{1}(H)$ (see [30] for a
proof of the closedness.)

$\mathcal{P}_{R}:=\\{\mu\in\mathcal{P}(H):\int_{H}g(w)\mu(dw)\leq R\\},\quad
R>0.$

By the contraction property in Theorem 2.7, we have that for any $R>0$ and
$\mu\in\mathcal{P}_{R}$, $\varphi$ is continuous in $\mu$, uniformly with
respect to $h$. And by inequality (3.14), it is continuous in $h$ uniformly
for $\mu$. Hence $\varphi$ is joint continuous in
$(\mu,h)\in\mathcal{P}_{R}\times\mathbb{T}^{n}$. Then the fixed point argument
will be applied on the complete subset $C(\mathbb{T}^{n},\mathcal{P}_{R})$.
However the trade off for the continuity is the loss of the invariance of
$C(\mathbb{T}^{n},\mathcal{P}_{R})$ under $S^{t}$ uniformly for any $t\geq 0$.
Indeed, it follows from Proposition 3.1 that for $\mu\in\mathcal{P}_{R}$, and
any $t\geq 0$,

$\displaystyle\int_{H}\mathcal{P}_{0,t,h}g(w)\mu(dw)$
$\displaystyle=\int_{H}\mathbf{E}g(\mathrm{\Phi}_{0,t,h}(w))\mu(dw)$
$\displaystyle\leq C\int_{H}g^{\alpha(t)}(w)\mu(dw)\leq
C\left(\int_{H}g(w)\mu(dw)\right)^{\alpha(t)}\leq CR^{\alpha(t)},$

where we used Jensen’s inequality in the penultimate step. One can check that
it is impossible to choose a common $R>0$ such that $CR^{\alpha(t)}\leq R$ for
any $t\geq 0$ since $\alpha(t)\rightarrow 0$ as $t\rightarrow 0$. However,
note that for each fixed $t_{0}>0$, if we choose
$R=R_{t_{0}}:=C^{\frac{1}{1-\alpha(t_{0})}}$ then $CR^{\alpha(t)}\leq
CR^{\alpha(t_{0})}=R_{t_{0}}$, which gives the invariance under $S^{t}$
uniformly for $t\geq t_{0}$.

Now for any fixed $t_{0}\in(0,1)$, the above analysis shows that the map
$S^{t}:C(\mathbb{T}^{n},\mathcal{P}_{R_{t_{0}}})\rightarrow
C(\mathbb{T}^{n},\mathcal{P}_{R_{t_{0}}})$ is well defined for $t\geq t_{0}$.
It remains to show that it is a contraction. Indeed, by Theorem 2.7, one has

$\displaystyle p(S^{t}\gamma_{1},S^{t}\gamma_{2})$
$\displaystyle=\max_{h\in{\mathbb{T}^{n}}}\rho(\varphi(t,\gamma_{1}(\beta_{-t}h),\beta_{-t}h),\varphi(t,\gamma_{2}(\beta_{-t}h),\beta_{-t}h))$
$\displaystyle=\max_{h\in{\mathbb{T}^{n}}}\rho(\mathcal{P}_{0,t,\beta_{-t}h}^{*}\gamma_{1}(\beta_{-t}h),\mathcal{P}_{0,t,\beta_{-t}h}^{*}\gamma_{2}(\beta_{-t}h))$
$\displaystyle\leq Ce^{-\varpi
t}\max_{h\in{\mathbb{T}^{n}}}\rho(\gamma_{1}(\beta_{-t}h),\gamma_{2}(\beta_{-t}h))=Ce^{-\varpi
t}p(\gamma_{1},\gamma_{2}).$

Therefore for large $T>t_{0}$, $p(S^{T}\gamma_{1},S^{T}\gamma_{2})\leq
cp(\gamma_{1},\gamma_{2})$ for some $c\in(0,1)$. Fix such a $T$, then $S^{T}$
is a contraction over the complete metric space
$C({\mathbb{T}^{n}},\mathcal{P}_{R_{t_{0}}})$, so there is a unique fixed
point $\Gamma_{t_{0}}\in C({\mathbb{T}^{n}},\mathcal{P}_{R_{t_{0}}})$ of
$S^{T}$. Noting for any $t\geq t_{0}$, $S^{t}$ maps
$C({\mathbb{T}^{n}},\mathcal{P}_{R_{t_{0}}})$ to itself, hence

$S^{T}(S^{t}\Gamma_{t_{0}})=S^{t}(S^{T}\Gamma_{t_{0}})=S^{t}(\Gamma_{t_{0}})$

implies that $S^{t}(\Gamma_{t_{0}})=\Gamma_{t_{0}}$ by the uniqueness of the
fixed point, which shows that $\Gamma_{t_{0}}$ is a fixed point of $S^{t}$ for
$t\geq t_{0}$. For $0<t_{1}\leq t_{0}$, one has $R_{t_{1}}\geq R_{t_{0}}$, so
$\mathcal{P}_{R_{t_{0}}}\subset\mathcal{P}_{R_{t_{1}}}$. And for the same
$T>0$, $S^{T}$ is a contraction on
$C({\mathbb{T}^{n}},\mathcal{P}_{R_{t_{1}}})$, which has a unique fixed point
$\Gamma_{t_{1}}$. By the uniqueness, $\Gamma_{t_{1}}=\Gamma_{t_{0}}$, hence
$\Gamma_{t_{0}}$ is also a fixed point of $S^{t}$ for $t\geq t_{1}$. Since
$t_{1}$ is arbitrary, we see that $\Gamma:=\Gamma_{t_{0}}$ is a fixed point of
$S^{t}$ for $t\geq 0$, that is,
$\varphi(t,\Gamma(\beta_{-t}h),\beta_{-t}h)=\Gamma(h)$ for all
$h\in\mathbb{T}^{n}$. Replacing $h$ with $\beta_{t}h$ we have
$\varphi(t,\Gamma(h),h)=\Gamma(\beta_{t}u)$ which by definition is

$\mathcal{P}_{0,t,h}^{*}\Gamma(h)=\Gamma(\beta_{t}h).$

Hence the invariance follows, and the exponential mixing (3.13) then follows
from the invariance and Theorem 2.7. Note that by replacing $h$ with
$\beta_{s}h$ in the invariance identity and using the translation identity
(2.11), we have

(3.15)
$\displaystyle\mathcal{P}_{s,s+t,h}^{*}\Gamma(\beta_{s}h)=\Gamma(\beta_{s+t}h),\quad\forall
s\in\mathbb{R},t\geq 0,h\in\mathbb{T}^{n}.$

To show the uniqueness of $\Gamma$, suppose that there is another
$\widetilde{\Gamma}\in C(\mathbb{T}^{n},\mathcal{P}(H))$ that is invariant.
Then $\widetilde{\Gamma}\in C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$ by the
Lyapunov structure (3.1). Indeed, for $R>0$, let

$\displaystyle g_{R}(w)=\left\\{\begin{array}[]{rr}e^{2k\eta\|w\|^{2}},\text{
if }\|w\|\leq R,\\\ e^{2k\eta R^{2}},\text{ if }\|w\|\geq R.\\\
\end{array}\right.$

Then by the invariance of $\widetilde{\Gamma}$ and estimate (3.1), we have for
any $M,N>0$,

$\displaystyle\int_{H}g_{R}(w)\widetilde{\Gamma}_{h}(dw)$
$\displaystyle=\int_{H}\mathcal{P}_{-N,0,h}g_{R}(w)\widetilde{\Gamma}_{\beta_{-N}h}(dw)$
$\displaystyle\leq\int_{\\{\|w\|\leq
M\\}}\mathcal{P}_{-N,0,h}g_{R}(w)\widetilde{\Gamma}_{\beta_{-N}h}(dw)+\int_{\\{\|w\|\geq
M\\}}\mathcal{P}_{-N,0,h}g_{R}(w)\widetilde{\Gamma}_{\beta_{-N}h}(dw)$
$\displaystyle\leq\int_{\\{\|w\|\leq
M\\}}\mathbf{E}g(w_{-N,0,h}(w))\widetilde{\Gamma}_{\beta_{-N}h}(dw)+e^{2k\eta
R^{2}}\widetilde{\Gamma}_{\beta_{-N}h}(\\{\|w\|\geq M\\})$ $\displaystyle\leq
Ce^{2\kappa\eta\alpha(N)M^{2}}+e^{2\kappa\eta
R^{2}}\widetilde{\Gamma}_{\beta_{-N}h}(\\{\|w\|\geq M\\}).$

Since $\mathbb{T}^{n}$ is compact, and $\widetilde{\Gamma}\in
C(\mathbb{T}^{n},\mathcal{P}(H))$, where $\mathcal{P}(H)$ is endowed with the
topology of weak convergence, therefore
$\\{\widetilde{\Gamma}_{h}\\}_{h\in\mathbb{T}^{n}}$ is compact and hence tight
by Prokhorov’s theorem: for any $\varepsilon>0$, there is a compact subset
$K_{\varepsilon}$ of $H$ such that

$\widetilde{\Gamma}_{h}(H\backslash K_{\varepsilon})<\varepsilon,\quad\forall
h\in\mathbb{T}^{n}.$

Hence for any $R>0$ and $\varepsilon=e^{-2\kappa\eta R^{2}}$, there is a
compact subset $K_{\varepsilon}$ of $H$ such that

$\widetilde{\Gamma}_{\beta_{-N}h}(H\backslash
K_{\varepsilon})<\varepsilon,\quad\forall N>0.$

Now we can choose $M$ large enough such that
$K_{\varepsilon}\subset\\{\|w\|\leq M\\}$ so that

$e^{2\kappa\eta R^{2}}\widetilde{\Gamma}_{\beta_{-N}h}(\\{\|w\|\geq M\\})\leq
1.$

Since $\alpha(N)\rightarrow 0$ as $N\rightarrow\infty$, we can choose $N$
large such that $e^{2\kappa\eta\alpha(N)M^{2}}\leq 1$ as well. Therefore we
have

$\int_{H}g_{R}(w)\widetilde{\Gamma}_{h}(dw)\leq C,\quad\forall R>0,$

which, by the monotone convergence theorem, in turn implies that

$\int_{H}g(w)\widetilde{\Gamma}_{h}(dw)\leq C,\quad\forall
h\in\mathbb{T}^{n},$

and hence $\widetilde{\Gamma}\in C(\mathbb{T}^{n},\mathcal{P}_{1}(H))$. This
ensures that

$\sup_{h\in\mathbb{T}^{n}}\rho(\Gamma(h),\widetilde{\Gamma}(h))<\infty.$

Now by the translation identity (2.11) and Theorem 2.7, we have for
$h\in\mathbb{T}^{n},t\geq s$,

$\displaystyle\rho(\Gamma(\beta_{t}h),\widetilde{\Gamma}(\beta_{t}h))$
$\displaystyle=\rho(\mathcal{P}_{s,t,h}^{*}\Gamma(\beta_{s}h),\mathcal{P}_{s,t,h}^{*}\widetilde{\Gamma}(\beta_{s}h))$
$\displaystyle\leq
Ce^{-\varpi(t-s)}\rho(\Gamma(\beta_{s}h),\widetilde{\Gamma}(\beta_{s}h))\leq
C\sup_{h\in\mathbb{T}^{n}}\rho(\Gamma(h),\widetilde{\Gamma}(h))e^{-\varpi(t-s)}.$

By letting $s\rightarrow-\infty$, it follows that
$\Gamma(\beta_{t}h)=\widetilde{\Gamma}(\beta_{t}h)$ for $t\in\mathbb{R}$. It
particular this is true for $t=0$ and any $h\in\mathbb{T}^{n}$, hence
$\Gamma=\widetilde{\Gamma}$.

To show that $\Gamma\in C^{\zeta}(\mathbb{T}^{n},(\mathcal{P}_{1}(H),\rho))$
if $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$, where
$\zeta=\frac{\varpi\gamma}{r+\varpi}$ with
$r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ from estimate
(6.3), observing that for any $t\geq 0$ and $h_{1},h_{2}\in\mathbb{T}^{n}$, by
the invariance of $\Gamma$ and estimate (6.3), one has

$\displaystyle\rho\Big{(}\Gamma(h_{1}),\Gamma(h_{2})\Big{)}=\rho\Big{(}\varphi\left(t,\Gamma(\beta_{-t}h_{1}),\beta_{-t}h_{1}\right),\varphi\left(t,\Gamma(\beta_{-t}h_{2}),\beta_{-t}h_{2}\right)\Big{)}$
$\displaystyle\leq\rho\Big{(}\varphi\left(t,\Gamma(\beta_{-t}h_{1}),\beta_{-t}h_{1}\right),\varphi\left(t,\Gamma(\beta_{-t}h_{2}),\beta_{-t}h_{1}\right)\Big{)}+\rho\Big{(}\varphi\left(t,\Gamma(\beta_{-t}h_{2}),\beta_{-t}h_{1}\right),\varphi\left(t,\Gamma(\beta_{-t}h_{2}),\beta_{-t}h_{2}\right)\Big{)}$
$\displaystyle\leq
Ce^{rt}\int_{H}g(w)\Gamma(\beta_{-t}h_{1})(dw)\|\Psi\|_{\gamma}|h_{1}-h_{2}|^{\gamma}+Ce^{-\varpi
t}\rho\Big{(}\Gamma(\beta_{-t}h_{1}),\Gamma(\beta_{-t}h_{2})\Big{)}$
$\displaystyle\leq
Ce^{rt}R_{t_{0}}\|\Psi\|_{\gamma}|h_{1}-h_{2}|^{\gamma}+Ce^{-\varpi
t}\sup_{h_{1},h_{2}\in\mathbb{T}^{n}}\rho\Big{(}\Gamma(h_{1}),\Gamma(h_{2})\Big{)}$
$\displaystyle\leq C(e^{rt}|h_{1}-h_{2}|^{\gamma}+e^{-\varpi t})\leq
C|h_{1}-h_{2}|^{\zeta},$

with $\zeta=\frac{\varpi\gamma}{r+\varpi}$, by applying the following lemma.

######  Lemma 3.8.

For $D\geq 1,\Lambda_{1},\Lambda_{2}>0,\gamma\in(0,1],0<\delta\leq D$, one has

$\displaystyle e^{\Lambda_{1}T}\delta^{\gamma}+e^{-\Lambda_{2}T}\leq
2D^{\gamma}\delta^{\overline{\gamma}},$

for $\overline{\gamma}=\frac{\Lambda_{2}}{\Lambda_{1}+\Lambda_{2}}\gamma$, by
choosing $T=-\frac{\gamma}{\Lambda_{1}+\Lambda_{2}}\ln\delta$ for $\delta<1$
and $T=0$ for $\delta\geq 1$.

The proof is then complete. ∎

We have the following two implications that express the mixing property in
terms of the action of the transition operators on observables. Since the
proof is the same as that in [30], we only state the results here and omit the
proof.

######  Corollary 3.9.

There is a constant $\eta_{0}>0$, such that for every $\eta\in(0,\eta_{0}]$,
there exist constants $C,\varpi>0$ such that

$\|\mathcal{P}_{s,t,h}\phi-\Gamma_{\beta_{t}h}(\phi)\|_{\rho,h,s}\leq
Ce^{-\varpi(t-s)}\|\phi-\Gamma_{\beta_{s}h}(\phi)\|_{\rho,h,s},$

for every Fréchet differentiable $\phi:H\rightarrow\mathbb{R}$, $s\leq t$ and
$h\in\mathbb{T}^{n}$. Here

$\displaystyle\|\phi\|_{\rho,h,s}=\sup_{u\neq
v}\frac{|\phi(u)-\phi(v)|}{\rho(u,v)}+\left|\langle\Gamma_{\beta_{s}h},\phi\rangle\right|$

######  Theorem 3.10.

There is a constant $\eta_{0}>0$, such that for every $\eta\in(0,\eta_{0}]$,
there exist constants $C,\varpi>0$, such that for every $\phi\in C_{\eta}^{1}$
as in Theorem 2.6,

(3.16)
$\displaystyle\|\mathcal{P}_{0,t,h}\phi-\int_{H}\phi(w)\Gamma_{\beta_{t}h}(dw)\|_{\eta}\leq
Ce^{-\varpi t}\|\phi\|_{\eta}.$

for any $h\in\mathbb{T}^{n},t\geq 0$. In particular, the inequality (2.15) is
obtained by replacing $h$ with $\beta_{s}0$ and using the translation identity
(2.11), where $\mu_{s}:=\Gamma_{\beta_{s}0}$.

It turns out that Theorem 3.7 also implies the convergence of time averages of
the transition probabilities, which is quite useful when applied to the proof
of the limit theorems in the next subsection.

######  Proposition 3.11.

For any $(w_{0},h)\in H\times\mathbb{T}^{n}$ and $K\in\mathbb{N}$, we have the
following weak convergence of measures:

1\.
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\mathcal{P}^{*}_{0,(j-1)K,h}\delta_{w_{0}}\rightarrow\int_{\mathbb{T}^{n}}\Gamma_{g}\lambda(dg).$
2\.
$\displaystyle\frac{1}{N}\sum_{j=1}^{N}P^{*}_{(j-1)K}\delta_{(w_{0},h)}\rightarrow\Gamma_{g}(dw)\lambda(dg)$
and
$\displaystyle\frac{1}{T}\int_{0}^{T}P^{*}_{t}\delta_{(w_{0},h)}dt\rightarrow\Gamma_{g}(dw)\lambda(dg)$
as well.

###### Proof.

For any $\phi\in\mathrm{Lip}_{\rho}(H)$, we have by the Monge-Kantorovich
duality (2.8), the invariance and mixing of the quasi-periodic invariant
measure from Theorem 3.7 that

$\displaystyle\left|\left\langle\frac{1}{N}\sum_{j=1}^{N}\left(\mathcal{P}^{*}_{0,(j-1)K,h}\delta_{w_{0}}-\Gamma_{\beta_{(j-1)K}h}\right),\phi\right\rangle\right|$
$\displaystyle\leq\mathrm{Lip}_{\rho}(\phi)\frac{1}{N}\sum_{j=1}^{N}\rho({P}^{*}_{0,(j-1)K,h}\delta_{w_{0}},{P}^{*}_{0,(j-1)K,h}\Gamma_{h})\leq
C\mathrm{Lip}_{\rho}(\phi)\frac{1}{N}\sum_{j=1}^{N}e^{-\varpi(j-1)K}\rho(\delta_{0},\Gamma_{h}),$

which tends to $0$ as $N\rightarrow\infty$. Also by Birkhoff’s ergodic theorem
for the irrational rotation on $\mathbb{T}^{n}$, one has

$\left\langle\frac{1}{N}\sum_{j=1}^{N}\Gamma_{\beta_{(j-1)K}h},\phi\right\rangle\rightarrow\int_{\mathbb{T}^{n}}\langle\Gamma_{g},\phi\rangle\lambda(dg).$

Hence the first claim of the lemma follows.

Now let $\phi\in\mathrm{Lip}_{\rho,d}(H\times\mathbb{T}^{n})$, where $H$ is
equipped with the metric $\rho$ and $d$ is the usual distance in
$\mathbb{T}^{n}$ induced from $\mathbb{R}^{n}$. Observe that

$\displaystyle\left\langle\frac{1}{N}\sum_{j=1}^{N}P^{*}_{(j-1)K}\delta_{(w_{0},h)},\phi\right\rangle=\frac{1}{N}\sum_{j=1}^{N}\mathcal{P}_{0,(j-1)K,h}\phi(\cdot,\beta_{(j-1)K}h)(w_{0})$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}\Big{\langle}\mathcal{P}_{0,(j-1)K,h}^{*}\delta_{w_{0}}-\mathcal{P}_{0,(j-1)K,h}^{*}\Gamma_{h},\phi(\cdot,\beta_{(j-1)K}h)\Big{\rangle}+\frac{1}{N}\sum_{j=1}^{N}\Big{\langle}\mathcal{P}_{0,(j-1)K,h}^{*}\Gamma_{h},\phi(\cdot,\beta_{(j-1)K}h)\Big{\rangle}$
$\displaystyle:=I+II,$

where the first term in the sum vanishes by the mixing of the quasi-periodic
invariant measure since

$\displaystyle|I|\leq\frac{1}{N}\sum_{j=1}^{N}\mathrm{Lip}_{\rho}\phi(\cdot,\sigma_{(j-1)K}h)\rho(\mathcal{P}_{0,(j-1)K,h}^{*}\delta_{w},\mathcal{P}_{0,(j-1)K,h}^{*}\Gamma_{h})\leq\frac{C\mathrm{Lip}_{\rho,d}(\phi)}{N}\sum_{j=1}^{N}e^{-\varpi(j-1)K}\rho(\delta_{w_{0}},\Gamma_{h})\rightarrow
0.$

While the second term converges to the average of $\phi$ with respect to
$\Gamma_{h}(dw)\lambda(dg)$ by Birkhoff’s ergodic theorem for the irrational
rotation $\beta$:

$\displaystyle
II=\frac{1}{N}\sum_{j=1}^{N}\Big{\langle}\Gamma_{\sigma_{(j-1)K}h},\phi(\cdot,\beta_{(j-1)K}h)\Big{\rangle}\rightarrow\int_{H\times\mathbb{T}^{n}}\phi(w,g)\Gamma_{g}(dw)\lambda(dg),$

since the observable
$g\in\mathbb{T}^{n}\rightarrow\Big{\langle}\Gamma_{g},\phi(\cdot,g)\Big{\rangle}$
is continuous. The proof for the continuous time version is similar. ∎

######  Remark 3.1.

Usually the sequence of time averages of the transition probabilities always
possesses a subsequence that converges to an invariant measure by the Krylov-
Bogoliubov theorem, however it is not guaranteed that the whole sequence
always converges to the invariant measure. While in the above lemma, even if
it is the case that the homogenized Markov process is not mixing, we have the
convergence of the time averages to the unique ergodic invariant measure. From
the proof we see that this is a result of the mixing of the inhomogeneous
Markov process along the $H$ component, together with the unique ergodicity of
the irrational rotation flow.

##  4\. SLLN and CLT for time inhomogeneous solution process

In this section, we establish the limit theorems as well as the corresponding
convergence rates as given in Theorem 2.8 and Theorem 2.9 for the time
inhomogeneous solution process of the Navier-Stokes equation (2.3). In fact,
we will prove these results for more general class of observable functions
that will be given below. The proof is based on a martingale approximation and
the limit theorems from the martingale theory.

Due to the time inhomogeneity, it is not obvious to derive a martingale
approximation for the inhomogeneous solution process. We also note that the
homogenized process $X_{t}$ is not mixing in the usual sense since
$\mathbf{E}\phi(X_{t})-\int_{H\times\mathbb{T}^{n}}\phi(w,h)\Gamma_{h}(dw)\lambda(dh)$
does not decay to $0$, essentially because the irrational rotation on the
torus is not mixing. Here $\Gamma_{h}(dw)\lambda(dh)$ is the unique invariant
measure for $X_{t}$. Therefore the usual martingale approximation cannot be
applied directly. Yet the exponentially mixing quasi-periodic invariant
measure enables us to center the observation along the solution process in an
appropriate way, which gives us a chance to have a martingale approximation.
Indeed, by mixing (3.7), the transition probabilities are exponentially
attracted by the quasi-periodic invariant measure, so the expectation of the
observation $\phi(w_{0,t}(w_{0}))$ along the inhomogeneous process is
attracted by the quasi-periodic path $\int_{H}\phi(w)\mu_{t}(dw)$. Hence
$\phi(w_{0,t}(w_{0}))-\int_{H}\phi(w)\mu_{t}(dw)$ forms a family of
“asymptotically centered” random variables that is expected to have the
asymptotic behavior described by the limit theorems. Note that if we let
$\widetilde{\phi}(w,h)=\phi(w)-\langle\phi,\Gamma_{h}\rangle$, then

$\phi(w_{0,t}(w_{0}))-\int_{H}\phi(w)\mu_{t}(dw)=\widetilde{\phi}(X_{t}(w_{0},0)),$

which is the observation of the centered observable function along the
homogenized process $X_{t}(w_{0},0)$. This indicates that the homogenized
process is mixing when acting on the observables centered by the quasi-
periodic invariant measure, which enables us to derive a martingale
approximation.

Subsection 4.1 below is devoted to the study of this particular martingale
approximation. Then the strong law of large numbers and the central limit
theorem are direct conclusions of the corresponding theorems established in
martingale theory, which merely requires the weak convergence of the average
of the transition probabilities established in Proposition 3.11 and the bounds
on the approximating martingale. The proof is presented in subsection 4.2. The
rate of convergence for the limit theorems will be given in subsection 4.3,
which requires more detailed analyses on regularity of observables and the
convergence rate for the moments of the time average of the observations
centered by the quasi-periodic invariant measure.

We now define the space of observable functions and state the main results of
this section. For $\gamma\in(0,1]$, let
$C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$ be the space of Hölder continuous
functions with finite norms weighted by the Lyapunov function
$e^{\eta\|w\|^{2}}$,

(4.1) $\displaystyle
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n}):=\left\\{\phi\in
C(H\times\mathbb{T}^{n}):\|\phi\|_{\gamma,\eta,H}<\infty\right\\},$

where

(4.2) $\displaystyle\|\phi\|_{\gamma,\eta,H}:=\sup_{(w,h)\in
H\times\mathbb{T}^{n}}\frac{|\phi(w,h)|}{e^{\eta\|w\|^{2}}}+\sup_{\begin{subarray}{c}h\in\mathbb{T}^{n}\\\
0<\|w_{1}-w_{2}\|\leq
1\end{subarray}}\frac{|\phi(w_{1},h)-\phi(w_{2},h)|}{\|w_{1}-w_{2}\|^{\gamma}\left(e^{\eta\|w_{1}\|^{2}}+e^{\eta\|w_{2}\|^{2}}\right)}.$

Let also $C_{\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$ be the
space of functions that are Hölder continuous on $\mathbb{T}^{n}-$component
and uniformly on bounded subset of $H$

(4.3) $\displaystyle
C_{\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n}):=\left\\{\phi\in
C(H\times\mathbb{T}^{n}):\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}<\infty\right\\},$

where

$\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}:=\sup_{(w,h)\in
H\times\mathbb{T}^{n}}\frac{|\phi(w,h)|}{e^{\eta\|w\|^{2}}}+\sup_{\begin{subarray}{c}w\in
H\\\ 0<\|h_{1}-h_{2}\|\leq
1\end{subarray}}\frac{|\phi(w,h_{1})-\phi(w,h_{2})|}{e^{\eta\|w\|^{2}}|h_{1}-h_{2}|^{\gamma}}.$

Recall that $\rho$ is the geodesic distance on $H$ weighted by
$e^{\eta\|w\|^{2}}$ and therefore depends on $\eta$. Let

$[\phi]_{\gamma,\eta,H_{\rho}}=\sup_{\begin{subarray}{c}h\in\mathbb{T}^{n}\\\
0<\rho(w_{1},w_{2})\leq
1\end{subarray}}\frac{|\phi(w_{1},h)-\phi(w_{2},h)|}{\rho(w_{1},w_{2})^{\gamma}},$

be the Hölder semi-norm under the metric $\rho$, and set
$C^{\gamma}_{\eta,H_{\rho}}(H\times\mathbb{T}^{n})$ as the space of bounded
Hölder continuous (with respect to the metric $\rho$ on $H$) functions

(4.4) $\displaystyle
C^{\gamma}_{\eta,H_{\rho}}(H\times\mathbb{T}^{n})=\left\\{\phi\in
C(H\times\mathbb{T}^{n}):\|\phi\|_{\gamma,\eta,H_{\rho}}:=\sup_{(w,h)\in
H\times\mathbb{T}^{n}}|\phi(w,h)|+[\phi]_{\gamma,\eta,H_{\rho}}<\infty\right\\}.$

######  Remark 4.1.

It is straightforward to verify that for $0<\delta\leq\gamma$, the following
inclusion holds:

$\left\\{\phi\in
C(H\times\mathbb{T}^{n}):[\phi]_{\gamma,\eta,H_{\rho}}<\infty\right\\}\subset
C^{\delta}_{2\eta,H}(H\times\mathbb{T}^{n}).$

In the case when the deterministic force $f(t,x)$ vanishes and the noise is
degenerate as in our work, the weak law of large numbers and central limit
theorem were proved in [29] for Lipschitz observable functions $\phi$ with
$[\phi]_{1,\eta,H_{\rho}}<\infty.$ In view of the above inclusion and the fact
that the following Theorem 4.1 and Theorem 4.2 are valid for $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, our limit theorems in the case
when the deterministic force $f=0$ can be considered as an improvement of
those in [29].

For any $\phi\in C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, we set
$\widetilde{\phi}$ as the associated function obtained by normalizing $\phi$
with the quasi-periodic invariant measure,

(4.5)
$\displaystyle\widetilde{\phi}(w,h)=\phi(w,h)-\langle\Gamma_{h},\phi(\cdot,h)\rangle.$

It is clear that $\widetilde{\phi}\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$ as well. Note that
$X_{t}(w_{0},h_{0})=(w_{0,t,h_{0}}(w_{0}),\beta_{t}h_{0})$, hence

$\displaystyle\widetilde{\phi}(X_{t}(w_{0},h_{0}))=\phi(w_{0,t,h_{0}}(w_{0}),\beta_{t}h_{0})-\langle\Gamma_{\beta_{t}h_{0}},\phi(\cdot,\beta_{t}h_{0})\rangle.$

Recall from Theorem 3.7 that when $h_{0}=0$, $\Gamma_{\beta_{t}0}=\mu_{t}$ is
the unique quasi-periodic invariant measure of the Navier-Stokes system (2.3)
with the deterministic force $f(t,x)$. And when $\phi$ is an observable
function on $H$, we have
$\widetilde{\phi}(X_{t}(w_{0},0))=\phi(w_{0,t,h_{0}}(w_{0}))-\langle\mu_{t},\phi\rangle$,
which is the observation along the solution process normalized by the quasi-
periodic invariant measure. In particular, Theorem 2.8 and Theorem 2.9 with
$s=0$ are obtained from the following Theorem 4.1-4.4 by taking the observable
function $\phi\in C_{\eta}^{\gamma}(H)\subset
C_{\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$ and $h_{0}=0$. For simplicity, we
prove Theorem 4.1-4.4 for initial time $s=0$, while the proof applies to
$s\neq 0$ without any change. Therefore Theorem 2.8 and Theorem 2.9 hold for
any initial time $s\in\mathbb{R}$.

Let $\eta_{0}$ be the constant from (6.1). The first result is the strong law
of large numbers with it’s rate of convergence.

######  Theorem 4.1 (SLLN).

For any $\eta\in(0,2^{-4}\eta_{0}]$, $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, $(w_{0},h_{0})\in
H\times\mathbb{T}^{n}$, and $\varepsilon>0$,

(4.6)
$\displaystyle\lim_{T\rightarrow\infty}T^{-\frac{1}{2}-\varepsilon}\int_{0}^{T}\widetilde{\phi}(X_{t}(w_{0},h_{0}))dt=0,\quad\mathbf{P}\text{-}\mathrm{a.s}..$

The second result is the central limit theorem.

######  Theorem 4.2 (CLT).

For any $\eta\in(0,2^{-6}\eta_{0}]$, $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, and $(w_{0},h_{0})\in
H\times\mathbb{T}^{n}$, one has

(4.7)
$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{\sqrt{T}}\int_{0}^{T}\widetilde{\phi}(X_{t}(w_{0},h_{0}))dt\overset{\mathcal{D}}{=}N(0,\sigma^{2}),$

where $N(0,\sigma^{2})$ is the standard normal random variable with variance
$\sigma^{2}$ and $\mathcal{D}$ represents the convergence in distribution. In
fact

$\displaystyle\sigma^{2}=\sigma_{\phi}^{2}=\lim_{T\rightarrow\infty}\frac{1}{T}\mathbf{E}\left(\int_{0}^{T}\widetilde{\phi}\left(X_{t}(w_{0},h_{0})\right)dt\right)^{2}.$

The third result of this section is an estimate of the rate of convergence for
the strong law of large numbers.

######  Theorem 4.3.

Let $\varepsilon>0$, for every integer $p\geq 3$ satisfying
$2^{p}>1/\varepsilon$, every $\eta\in(0,2^{-p-1}\eta_{0}]$, and every $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, $(w_{0},h_{0})\in
H\times\mathbb{T}^{n}$, there is an almost surely finite random time
$T_{0}(\omega)\geq 1$, depending on
$p,\varepsilon,\|\phi\|_{\gamma,\eta,H},\|w_{0}\|,h_{0}$ such that for all
$T>T_{0}$, we have

$\displaystyle\left|\frac{1}{T}\int_{0}^{T}\Big{(}\phi(X_{t}(w_{0},h_{0}))-\big{\langle}\Gamma_{\beta_{t}h},\phi(\cdot,\beta_{t}h)\big{\rangle}\Big{)}dt\right|\leq
CT^{-\frac{1}{2}+\varepsilon},$

where $C>0$ is a constant that does not depend on the above parameters.
Moreover, for every $0<\ell<\min\\{2^{p}\varepsilon-1,2^{p-2}-1\\}$, there is
a constant $C_{p}=C_{p}(\|\phi\|_{\gamma,\eta,H},\ell,\varepsilon)$ such that

$\mathbf{E}T_{0}^{\ell}\leq C_{p}e^{2^{p+1}\eta\|w_{0}\|^{2}}.$

The last result is on the convergence rate of the central limit theorem.

######  Theorem 4.4.

Assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$ and the frequency $\alpha$
satisfies the Diophantine condition (2.2) with constant $A$ and dimension $n$.
Let $\Lambda=\frac{\varpi}{5(2-\gamma)}$,
$\zeta=\frac{\varpi\gamma}{r+\varpi}$ and
$\overline{\gamma}_{0}=\frac{\Lambda\zeta}{5(\Lambda+r)(2-\gamma)}$, where
$\varpi$ is the mixing rate from Theorem 2.6 and
$r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ is the constant
from (6.3).

1\. For any integer $p\geq 2$, $\eta\in(0,2^{-p-1}\eta_{0}]$, and $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$ with $\sigma_{\phi}^{2}>0$, and
$(w_{0},h_{0})\in H\times\mathbb{T}^{n}$, there are constants
$C_{p}=C_{p}(\|\phi\|_{\gamma,\eta,H},\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}},\|w_{0}\|)>0$
and $T_{0}>0$ such that for all $T\geq T_{0}$,

$\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left\\{\frac{1}{\sqrt{T}}\int_{0}^{T}\widetilde{\phi}(X_{t}(w_{0},h_{0}))dt\leq
z\right\\}-\Phi_{\sigma_{\phi}}(z)\right|\leq
C_{p}\left(T^{-\frac{1}{4}}+T^{-\frac{2^{p-2}}{2^{p}+1}}+T^{-\frac{2^{p-1}\overline{\gamma}_{0}}{(2^{p}+1)(A+n)}}\right),$

2\. For $\eta\in(0,2^{-7}\eta_{0}]$ and $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$ such that $\sigma_{\phi}^{2}=0$,
and $(w_{0},h_{0})\in H\times\mathbb{T}^{n}$, there is a constant
$C=C(\|\phi\|_{\gamma,\eta,H},\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}},\|w_{0}\|)>0$
such that for all $T\geq 1$,

$\displaystyle\sup_{z\in\mathbb{R}}\left(|z|\wedge
1\right)\left|\mathbf{P}\left\\{\frac{1}{\sqrt{T}}\int_{0}^{T}\widetilde{\phi}(X_{t}(w_{0},h_{0}))dt\leq
z\right\\}-\Phi_{0}(z)\right|\leq
C\left(T^{-\frac{1}{4}}+T^{-\frac{\overline{\gamma}_{0}}{2(A+n)}}\right)$

###  4.1. The martingale approximation

We first give several properties of the spaces of observable functions defined
above in the following Proposition 4.5. Then we prove a mixing result (as a
consequence of Theorem 3.7) in terms of the observable functions in Theorem
4.6, which is crucial in deriving the martingale approximation. Proposition
4.7 gives the definition of the corrector and its properties that will be used
to construct the martingale approximation as given in (4.14).

######  Proposition 4.5.

Let $\eta_{0}$ be the constant from estimate (6.1), and $c_{0}$ be the
constant from the Ladyzhenskaya’s inequality (2.20). Then for
$\eta\in(0,\eta_{0}/2]$, and any $0<\gamma\leq 1$, $P_{t}$ maps
$C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$ into
$C^{\gamma}_{2\eta,H}(H\times\mathbb{T}^{n})$; If we further assume that
$\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$, then $P_{t}$ maps
$C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})\cap
C_{\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$ into
$C^{\gamma}_{2\eta,H}(H\times\mathbb{T}^{n})\cap
C_{2\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$.

###### Proof.

Let $\phi\in C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$. It follows from
(6.1) and (6.11) that for $\eta\in(0,\eta_{0}/2]$,

$\displaystyle|P_{t}\phi(w_{1},h)-P_{t}\phi(w_{2},h)|$
$\displaystyle=\left|\mathbf{E}\phi(w_{0,t,h}(w_{1}),\beta_{t}h)-\mathbf{E}\phi(w_{0,t,h}(w_{2}),\beta_{t}h)\right|\leq\mathbf{E}\left|\phi(w_{0,t,h}(w_{1}),\beta_{t}h)-\phi(w_{0,t,h}(w_{2}),\beta_{t}h)\right|$
$\displaystyle\leq\|\phi\|_{\gamma,\eta,H}\mathbf{E}\|w_{0,t,h}(w_{1})-w_{0,t,h}(w_{2})\|^{\gamma}\left(e^{\eta\|w_{0,t,h}(w_{1})\|^{2}}+e^{\eta\|w_{0,t,h}(w_{2})\|^{2}}\right)$
$\displaystyle\leq
C\|\phi\|_{\gamma,\eta,H}\left(\mathbf{E}\|w_{0,t,h}(w_{1})-w_{0,t,h}(w_{2})\|^{2}\right)^{\frac{\gamma}{2}}\left(\mathbf{E}\left(e^{2\eta\|w_{0,t,h}(w_{1})\|^{2}}+e^{2\eta\|w_{0,t,h}(w_{2})\|^{2}}\right)\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\|\phi\|_{\gamma,\eta,H}\|w_{1}-w_{2}\|^{\gamma}e^{\frac{\gamma
r}{2}t}e^{\frac{\gamma}{2}\eta\|w_{1}\|^{2}}\left(e^{\eta\|w_{1}\|^{2}}+e^{\eta\|w_{2}\|^{2}}\right),$

where $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C\left(f,\mathcal{B}_{0}\right)$ is
the constant from (6.4). Hence we have

(4.8) $\displaystyle|P_{t}\phi(w_{1},h)-P_{t}\phi(w_{2},h)|\leq
C\|\phi\|_{\gamma,\eta,H}\|w_{1}-w_{2}\|^{\gamma}e^{\frac{\gamma
r}{2}t}\left(e^{2\eta\|w_{1}\|^{2}}+e^{2\eta\|w_{2}\|^{2}}\right),$

which shows that $P_{t}$ maps $C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$
into $C^{\gamma}_{2\eta,H}(H\times\mathbb{T}^{n})$.

Now assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$ and let $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})\cap
C_{\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$. Then

$\displaystyle|P_{t}\phi(w,h_{1})-P_{t}\phi(w,h_{2})|=\left|\mathbf{E}\phi(w_{0,t,h_{1}}(w),\beta_{t}h_{1})-\mathbf{E}\phi(w_{0,t,h_{2}}(w),\beta_{t}h_{2})\right|$
$\displaystyle\leq\mathbf{E}\left|\phi(w_{0,t,h_{1}}(w),\beta_{t}h_{1})-\phi(w_{0,t,h_{1}}(w),\beta_{t}h_{2})\right|+\mathbf{E}\left|\phi(w_{0,t,h_{1}}(w),\beta_{t}h_{2})-\phi(w_{0,t,h_{2}}(w),\beta_{t}h_{2})\right|$
$\displaystyle\leq\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}|h_{1}-h_{2}|^{\gamma}\mathbf{E}e^{\eta\|w_{0,t,h_{1}}(w)\|^{2}}+\|\phi\|_{\gamma,\eta,H}\mathbf{E}\|w_{0,t,h_{1}}(w)-w_{0,t,h_{2}}(w)\|^{\gamma}\left(e^{\eta\|w_{0,t,h_{1}}(w)\|^{2}}+e^{\eta\|w_{0,t,h_{2}}(w)\|^{2}}\right)$

It follows from (6.1) and (6.3) that

$\displaystyle\mathbf{E}\|w_{0,t,h_{1}}(w)-w_{0,t,h_{2}}(w)\|^{\gamma}\left(e^{\eta\|w_{0,t,h_{1}}(w)\|^{2}}+e^{\eta\|w_{0,t,h_{2}}(w)\|^{2}}\right)$
$\displaystyle\leq\left(\mathbf{E}\|w_{0,t,h_{1}}(w)-w_{0,t,h_{2}}(w)\|^{2}\right)^{\frac{\gamma}{2}}\left(\mathbf{E}\left(e^{2\eta\|w_{0,t,h_{1}}(w)\|^{2}}+e^{2\eta\|w_{0,t,h_{2}}(w)\|^{2}}\right)\right)^{\frac{1}{2}}$
$\displaystyle\leq
Ce^{\frac{r\gamma}{2}t}e^{2\eta\|w\|^{2}}\|\Psi\|_{\gamma}^{\gamma}|h_{1}-h_{2}|^{\gamma}.$

Hence

(4.9) $\displaystyle|P_{t}\phi(w,h_{1})-P_{t}\phi(w,h_{2})|\leq
C(\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}+\|\phi\|_{\gamma,\eta,H}\|\Psi\|_{\gamma}^{\gamma})e^{\frac{r\gamma}{2}t}e^{2\eta\|w\|^{2}}|h_{1}-h_{2}|^{\gamma},$

which shows that $P_{t}$ maps $C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})\cap
C_{\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$ into
$C^{\gamma}_{2\eta,H}(H\times\mathbb{T}^{n})\cap
C_{2\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$. ∎

The following theorem shows that the homogenized process is mixing over the
family of observables normalized by the quasi-periodic invariant measure as in
(4.5), although it is not mixing in the usual sense (by centering the
observables with the unique ergodic invariant measure of the homogenized
process).

######  Theorem 4.6.

For any $\gamma\in(0,1]$, $\kappa\geq 2$, $\eta\in(0,\eta_{0}/\kappa]$ and
$\phi\in C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$ we have

(4.10) $\displaystyle|P_{t}\widetilde{\phi}(w,h)|\leq
C\|\phi\|_{\gamma,\eta,H}e^{\kappa\eta\|w\|^{2}}e^{-\Lambda t},\quad\forall
t\geq 0,(w,h)\in H\times\mathbb{T}^{n},$

where $\Lambda=\frac{\varpi}{5(2-\gamma)}$, the mixing rate $\varpi$ is from
Theorem 3.7, and $C$ is a positive constant independent of $(w,h)$ and
$\eta_{0}$ is the constant from (6.1).

###### Proof.

For any $R>0$, let $\chi_{R}:H\rightarrow\mathbb{R}\in
C_{\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$ satisfying $0\leq\chi_{R}\leq 1$,
with $\chi_{R}(w)=1$ for $\|w\|\leq R$ and $\chi_{R}(w)=0$ for $\|w\|\geq
R+1$. We can actually choose a $\chi_{R}$ such that
$\|\chi_{R}\|_{\gamma,\eta,H}\leq 2$. Assume without loss of generality that
$\|\phi\|_{\gamma,\eta,H}=1$. Also denote by $\overline{\chi}_{R}=1-\chi_{R}$.
Then

$\displaystyle|P_{t}\widetilde{\phi}(w,h)|$
$\displaystyle=|P_{t}\left(\chi_{R}\phi+\overline{\chi}_{R}\phi\right)(w,h)-\langle\Gamma_{\beta_{t}h},(\chi_{R}\phi)(\cdot,\beta_{t}h)+(\overline{\chi}_{R}\phi)(\cdot,\beta_{t}h)\rangle|$
$\displaystyle\leq|P_{t}(\chi_{R}\phi)(w,h)-\langle\Gamma_{\beta_{t}h},(\chi_{R}\phi)(\cdot,\beta_{t}h)\rangle|+|P_{t}(\overline{\chi}_{R}\phi)(w,h)-\langle\Gamma_{\beta_{t}h},(\overline{\chi}_{R}\phi)(\cdot,\beta_{t}h)\rangle|$
(4.11) $\displaystyle:=I_{1}+I_{2}.$

It is straightforward to show that $\chi_{R}\phi\in
C^{\gamma}_{\eta,H_{\rho}}(H\times\mathbb{T}^{n})$. Indeed, since
$\chi_{R}\phi$ vanishes outside of the ball $\|w\|\leq R+1$, in view of the
definition (4.2), one has

(4.12) $\displaystyle\sup_{(w,h)\in
H\times\mathbb{T}^{n}}|\chi_{R}(w)\phi(w,h)|\leq\sup_{\begin{subarray}{c}(w,h)\in
H\times\mathbb{T}^{n}\\\ \|w\|\leq
R+1\end{subarray}}|\phi(w,h)|\leq\|\phi\|_{\gamma,\eta,H}e^{\eta(R+1)^{2}}.$

Let

$S=\\{(w_{1},w_{2})\in H^{2}:\|w_{1}\|\leq R+1,\|w_{2}\|\geq
R+1,0<\|w_{1}-w_{2}\|\leq 1\\},$

and

$S_{\rho}=\\{(w_{1},w_{2})\in H^{2}:\|w_{1}\|\leq R+1,\|w_{2}\|\geq
R+1,0<\rho(w_{1},w_{2})\leq 1\\}.$

It is clear that $S_{\rho}\subset S$ since
$\|w_{1}-w_{2}\|\leq\rho(w_{1},w_{2})$. It follows from $\chi_{R}(w_{2})=0$,
$\|w_{2}\|\leq R+2$ and (4.12) that

$\displaystyle\sup_{(w_{1},w_{2})\in
S_{\rho}}\frac{|\chi_{R}(w_{1})\phi(w_{1},h)-\chi_{R}(w_{2})\phi(w_{2},h)|}{\rho(w_{1},w_{2})^{\gamma}}$
$\displaystyle=\sup_{(w_{1},w_{2})\in
S_{\rho}}\frac{|\chi_{R}(w_{1})\phi(w_{1},h)-\chi_{R}(w_{2})\phi(w_{1},h)|}{\rho(w_{1},w_{2})^{\gamma}}$
$\displaystyle\leq 2e^{\eta(R+1)^{2}}\sup_{(w_{1},w_{2})\in
S}\frac{|\chi_{R}(w_{1})-\chi_{R}(w_{2})|}{\|w_{1}-w_{2}\|^{\gamma}}$
$\displaystyle\leq 4e^{2\eta(R+2)^{2}}.$

Now let

$S^{R}=\\{(w_{1},w_{2})\in H^{2}:\|w_{1}\|,\|w_{2}\|\leq
R+1,0<\|w_{1}-w_{2}\|\leq 1\\},$

and

$S_{\rho}^{R}=\\{(w_{1},w_{2})\in H^{2}:\|w_{1}\|,\|w_{2}\|\leq
R+1,0<\rho(w_{1},w_{2})\leq 1\\}.$

First note that for $\|w_{1}\|,\|w_{2}\|\leq R+1$,

$\displaystyle|\chi_{R}(w_{1})\phi(w_{1},h)-\chi_{R}(w_{2})\phi(w_{2},h)|\leq
2e^{\eta(R+1)^{2}}\Big{(}|\chi_{R}(w_{1})-\chi_{R}(w_{2})+|\phi(w_{1},h)-\phi(w_{2},h)|\Big{)}$

Hence

$\displaystyle\sup_{(w_{1},w_{2})\in
S_{\rho}^{R}}\frac{|\chi_{R}(w_{1})\phi(w_{1},h)-\chi_{R}(w_{2})\phi(w_{2},h)|}{\rho(w_{1},w_{2})^{\gamma}}$
$\displaystyle\leq 2e^{\eta(R+1)^{2}}\Big{(}\sup_{(w_{1},w_{2})\in
S^{R}}\frac{|\chi_{R}(w_{1})-\chi_{R}(w_{2})|}{\|w_{1}-w_{2}\|^{\gamma}}+\sup_{(w_{1},w_{2})\in
S^{R}}\frac{|\phi(w_{1},h)-\phi(w_{2},h)|}{\|w_{1}-w_{2}\|^{\gamma}}\Big{)}$
$\displaystyle\leq 4e^{2\eta(R+2)^{2}}.$

It then follows that $\chi_{R}\phi\in
C^{\gamma}_{\eta,H_{\rho}}(H\times\mathbb{T}^{n})$ and

$\|\chi_{R}\phi\|_{\gamma,\eta,H_{\rho}}\leq Ce^{2\eta(R+2)^{2}},$

where $C$ is a constant that does not depend on $R$.

It is known that the dual Hölder metric on $\mathcal{P}(H)$ is bounded by the
Wasserstein metric (see Proposition 1.2.6 in [28] for example):

(4.13) $\displaystyle\sup_{\varphi\in
C_{b}^{\gamma}(H),\|\varphi\|_{\gamma}\leq
1}|\langle\mu_{1},\varphi\rangle-\langle\mu_{2},\varphi\rangle|\leq
5(\rho(\mu_{1},\mu_{2}))^{\frac{1}{2-\gamma}},\quad\forall\mu_{1},\mu_{2}\in\mathcal{P}(H),$

where $C_{b}^{\gamma}(H)$ is the space of bounded $\gamma-$Hölder continuous
functions on $H$ endowed with the metric $\rho$, and

$\|\varphi\|_{\gamma}=\sup_{w\in H}|\varphi(w)|+\sup_{0<\rho(w_{1},w_{2})\leq
1}\frac{|\varphi(w_{1})-\varphi(w_{2})|}{\rho(w_{1},w_{2})^{\gamma}}.$

Combining this fact with Theorem 3.7, it follows that the first term in (4.11)
satisfies

$\displaystyle I_{1}$
$\displaystyle=|P_{t}(\chi_{R}\phi)(w,h)-\langle\Gamma_{\beta_{t}h},(\chi_{R}\phi)(\cdot,\beta_{t}h)\rangle|$
$\displaystyle=|\langle\mathcal{P}_{0,t,h}^{*}\delta_{w},(\chi_{R}\phi)(\cdot,\beta_{t}h)\rangle-\langle\mathcal{P}_{0,t,h}^{*}\Gamma_{h},(\chi_{R}\phi)(\cdot,\beta_{t}h)\rangle|$
$\displaystyle\leq
5\|\chi_{R}\phi\|_{\gamma,\eta,H_{\rho}}(\rho(\mathcal{P}_{0,t,h}^{*}\delta_{w},\mathcal{P}_{0,t,h}^{*}\Gamma_{h}))^{\delta_{0}}\leq
Ce^{2\eta(R+2)^{2}}e^{-\delta_{0}\varpi t}e^{\kappa\eta\|w\|^{2}}.$

Here $\delta_{0}=\frac{1}{2-\gamma}\leq 1,\kappa\geq 2$ and $\varpi$ are
positive constants.

To estimate the second term in (4.11), observe that

$\displaystyle|P_{t}(\overline{\chi}_{R}\phi)(w,h)|$
$\displaystyle\leq\mathbf{E}_{(w,h)}(|\overline{\chi}_{R}\phi|)(X_{t})\leq(\mathbf{E}_{(w,h)}\overline{\chi}_{R}(X_{t}))^{1/2}(\mathbf{E}_{(w,h)}|\phi(X_{t})|^{2})^{1/2}$
$\displaystyle\leq\Big{(}\mathbf{P}\left(\|w_{0,t,h}(w)\|\geq
R\right)\Big{)}^{1/2}\Big{(}\mathbf{E}e^{2\eta\|w_{0,t,h}(w)\|^{2}}\Big{)}^{1/2}$
$\displaystyle\leq Ce^{\eta\|w\|^{2}}e^{-\eta
R^{2}}(\mathbf{E}e^{2\eta\|w_{0,t,h}(w)\|^{2}})^{1/2}\leq
Ce^{2\eta\|w\|^{2}}e^{-\eta R^{2}},$

where we used the fact that $\|\phi\|_{\gamma,\eta,H}=1$ and the estimate
(6.1) with a smaller $\eta_{0}$ (which is the one in (6.1) divided by $2$).

In a similar fashion, note that

$\displaystyle|\langle\Gamma_{\beta_{t}h},(\overline{\chi}_{R}\phi)(\cdot,\beta_{t}h)\rangle|$
$\displaystyle\leq\Big{(}\int_{H}\overline{\chi}_{R}(w)\Gamma_{\beta_{t}h}(dw)\Big{)}^{1/2}\Big{(}\int_{H}\left|\phi(w,\beta_{t}h)\right|^{2}\Gamma_{\beta_{t}h}(dw)\Big{)}^{1/2}$
$\displaystyle\leq\Big{(}\Gamma_{\beta_{t}h}(\|w\|\geq
R)\Big{)}^{1/2}\left(\int_{H}e^{2\eta\|w\|^{2}}\Gamma_{\beta_{t}h}(dw)\right)^{1/2}$
$\displaystyle\leq Ce^{-\eta
R^{2}}\left(\int_{H}e^{2\eta\|w\|^{2}}\Gamma_{\beta_{t}h}(dw)\right)^{1/2}\leq
Ce^{-\eta R^{2}}.$

As a result, we have the following estimate on (4.11),

$\displaystyle|P_{t}\widetilde{\phi}(w,h)|\leq
Ce^{\kappa\eta\|w\|^{2}}(e^{4\eta R^{2}-\delta_{0}\varpi t}+e^{-\eta R^{2}}).$

By choosing $R^{2}=\frac{\delta_{0}\varpi t}{5\eta}$, we obtain (4.10) with
$\Lambda=\frac{\delta_{0}\varpi}{5}=\frac{\varpi}{5(2-\gamma)}$. The proof is
complete.

∎

We now define the corrector that will be used in the martingale approximation
procedure.

######  Proposition 4.7 (The corrector).

For $\phi\in C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, define

$\chi(w,h):=\int_{0}^{\infty}P_{t}\widetilde{\phi}(w,h)dt=\int_{0}^{\infty}\mathcal{P}_{0,t,h}\phi(\cdot,\beta_{t}h)(w)-\langle\Gamma_{\beta_{t}h},\phi(\cdot,\beta_{t}h)\rangle
dt,\quad(w,h)\in H\times\mathbb{T}^{n}.$

(1). For $\eta\in(0,\eta_{0}/2]$, $\gamma\in(0,1]$, the corrector $\chi\in
C^{\gamma_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})$ as long as $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, where
$\gamma_{0}=\frac{\Lambda\gamma}{\Lambda+r}$, with $\Lambda$ from Theorem 4.6
and $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ is the constant
from (4.8).
(2). If we assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$ and $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\gamma}_{\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$, then the associated
corrector $\chi\in C^{\gamma_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\gamma_{1}}_{2\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$, where
$\gamma_{1}=\frac{\Lambda\zeta}{5(\Lambda+r)(2-\gamma)}$ with $\Lambda,r$ as
above and $\zeta$ from Theorem 3.7. In particular, we have

$\chi\in C^{\overline{\gamma}_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{2\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n}),$

where $\overline{\gamma}_{0}=\min\\{\gamma_{0},\gamma_{1}\\}=\gamma_{1}$.

###### Proof.

The function $\chi$ is well defined in view of Theorem 4.6. We begin with
proving the first item. Let $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$. It follows from Theorem 4.6 and
inequality (4.8) that, for $\eta\in(0,\eta_{0}/2]$,

$\displaystyle|\chi(w_{1},h)-\chi(w_{2},h)|$
$\displaystyle\leq\int_{0}^{T}|P_{t}\widetilde{\phi}(w_{1},h)-P_{t}\widetilde{\phi}(w_{2},h)|dt+\int_{T}^{\infty}|P_{t}\widetilde{\phi}(w_{1},h)|+|P_{t}\widetilde{\phi}(w_{2},h)|dt$
$\displaystyle\leq\int_{0}^{T}|P_{t}\phi(w_{1},h)-P_{t}\phi(w_{2},h)|dt+C\|\phi\|_{\gamma,\eta,H}e^{-\Lambda
T}\left(e^{2\eta\|w_{1}\|^{2}}+e^{2\eta\|w_{2}\|^{2}}\right)$
$\displaystyle\leq
C\|\phi\|_{\gamma,\eta,H}\left(\|w_{1}-w_{2}\|^{\gamma}e^{\frac{\gamma
r}{2}T}+e^{-\Lambda
T}\right)\left(e^{2\eta\|w_{1}\|^{2}}+e^{2\eta\|w_{2}\|^{2}}\right)$
$\displaystyle\leq
C\|\phi\|_{\gamma,\eta,H}\left(\|w_{1}-w_{2}\|^{\gamma}e^{rT}+e^{-\Lambda
T}\right)\left(e^{2\eta\|w_{1}\|^{2}}+e^{2\eta\|w_{2}\|^{2}}\right)$

where $r$ is from inequality (4.8). In view of Lemma 3.8, we have for any
$0<\|w_{1}-w_{2}\|\leq 1$,

$\displaystyle|\chi(w_{1},h)-\chi(w_{2},h)|\leq
C\|\phi\|_{\gamma,\eta,H}\left(e^{4\eta\|w_{1}\|^{2}}+e^{4\eta\|w_{2}\|^{2}}\right)\|w_{1}-w_{2}\|^{\gamma_{0}},$

with $\gamma_{0}=\frac{\Lambda\gamma}{\Lambda+r}$. This also indicates that
$\chi(w,h)$ is continuous in $w$ uniformly for $h$. The continuity of
$\chi(w,h)$ in $h$ for fixed $w$ follows from the fact that
$\chi_{T}(w,h):=\int_{0}^{T}P_{t}\widetilde{\phi}(w,h)dt$ is continuous in $h$
and $\chi_{T}(w,h)\rightarrow\chi(w,h)$ uniformly for $h$. Thus $\chi\in
C(H\times\mathbb{T}^{n})$. It also follows from Theorem 4.6 that
$|\chi(w,h)|\leq C\|\phi\|_{\gamma,\eta,H}e^{2\eta\|w\|^{2}}$. Hence $\chi\in
C^{\gamma_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})$ .

We now prove the second assertion. Let $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\gamma}_{\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$ and assume $\Psi\in
C^{\gamma}(\mathbb{T}^{n},H)$. Then

$\displaystyle|\chi(w,h_{1})-\chi(w,h_{2})|$
$\displaystyle\leq\int_{0}^{T}|P_{t}\widetilde{\phi}(w,h_{1})-P_{t}\widetilde{\phi}(w,h_{2})|dt+\int_{T}^{\infty}|P_{t}\widetilde{\phi}(w,h_{1})|+|P_{t}\widetilde{\phi}(w,h_{2})|dt$
$\displaystyle:=I+II.$

Note that by Theorem 4.6, the second term can be estimated as

$\displaystyle II\leq C\|\phi\|_{\gamma,\eta,H}e^{2\eta\|w\|^{2}}e^{-\Lambda
T}.$

For the first term, observe that

$\displaystyle I$
$\displaystyle\leq\int_{0}^{T}|P_{t}\phi(w,h_{1})-P_{t}\phi(w,h_{2})|dt+\int_{0}^{T}\left|\langle\Gamma_{\beta_{t}h_{1}},\phi(\cdot,\beta_{t}h_{1})\rangle-\langle\Gamma_{\beta_{t}h_{2}},\phi(\cdot,\beta_{t}h_{2})\rangle\right|dt$
$\displaystyle:=I_{1}+I_{2}.$

It then follows from the estimate (4.9) that

$\displaystyle I_{1}$ $\displaystyle\leq
C(\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}+\|\phi\|_{\gamma,\eta}\|\Psi\|_{\gamma}^{\gamma})e^{2\eta\|w\|^{2}}|h_{1}-h_{2}|^{\gamma}\int_{0}^{T}e^{\frac{r\gamma}{2}t}dt$
$\displaystyle\leq
C(\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}+\|\phi\|_{\gamma,\eta}\|\Psi\|_{\gamma}^{\gamma})e^{2\eta\|w\|^{2}}|h_{1}-h_{2}|^{\gamma}e^{rT}.$

To estimate $I_{2}$, noting that as in the proof of Theorem 4.6, especially by
(4.13) and Theorem 3.7 ($\zeta$-Hölder continuity of $\Gamma$), we have

$\displaystyle I_{21}:=$
$\displaystyle\left|\langle\Gamma_{\beta_{t}h_{1}},\phi(\cdot,\beta_{t}h_{1})\rangle-\langle\Gamma_{\beta_{t}h_{2}},\phi(\cdot,\beta_{t}h_{1})\rangle\right|$
$\displaystyle\leq\left|\langle\Gamma_{\beta_{t}h_{1}}-\Gamma_{\beta_{t}h_{2}},\chi_{R}\phi(\cdot,\beta_{t}h_{1})\rangle\right|+\left|\langle\Gamma_{\beta_{t}h_{1}},(\overline{\chi}_{R}\phi)(\cdot,\beta_{t}h_{1})\rangle\right|+\left|\langle\Gamma_{\beta_{t}h_{2}},(\overline{\chi}_{R}\phi)(\cdot,\beta_{t}h_{1})\rangle\right|$
$\displaystyle\leq C\|\phi\|_{\gamma,\eta,H}e^{4\eta
R^{2}}\|\Gamma\|_{\zeta}^{\frac{1}{2-\gamma}}|h_{1}-h_{2}|^{\frac{\zeta}{2-\gamma}}+C\|\phi\|_{\gamma,\eta,H}e^{-\eta
R^{2}}$ $\displaystyle\leq
C\|\phi\|_{\gamma,\eta,H}(\|\Gamma\|_{\zeta}^{\frac{1}{2-\gamma}}+1)\left(e^{4\eta
R^{2}}|h_{1}-h_{2}|^{\frac{\zeta}{2-\gamma}}+e^{-\eta R^{2}}\right)$
$\displaystyle\leq
C\|\phi\|_{\gamma,\eta,H}(\|\Gamma\|_{\zeta}^{\frac{1}{2-\gamma}}+1)|h_{1}-h_{2}|^{\frac{\zeta}{5(2-\gamma)}}$

by Lemma 3.8. Also note that by Theorem 3.7,

$\displaystyle I_{22}:=$
$\displaystyle\left|\langle\Gamma_{\beta_{t}h_{2}},\phi(\cdot,\beta_{t}h_{1})\rangle-\langle\Gamma_{\beta_{t}h_{2}},\phi(\cdot,\beta_{t}h_{2})\rangle\right|$
$\displaystyle\leq\langle\Gamma_{\beta_{t}h_{2}},|\phi(\cdot,\beta_{t}h_{1})-\phi(\cdot,\beta_{t}h_{2})|\rangle$
$\displaystyle\leq\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}|h_{1}-h_{2}|^{\gamma}\int_{H}e^{\eta\|w\|^{2}}\Gamma_{\beta_{t}h_{2}}(dw)\leq
C\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}|h_{1}-h_{2}|^{\gamma}.$

As a result,

$\displaystyle I_{2}\leq\int_{0}^{T}I_{21}+I_{22}dt\leq
CT\left(\|\phi\|_{\gamma,\eta,H}(\|\Gamma\|_{\zeta}^{\frac{1}{2-\gamma}}+1)+\|\phi\|_{\gamma,\eta,\mathbb{T}^{n}}\right)|h_{1}-h_{2}|^{\frac{\zeta}{5(2-\gamma)}}.$

Therefore by Lemma 3.8 again,

$\displaystyle|\chi(w,h_{1})-\chi(w,h_{2})|$ $\displaystyle\leq
Ce^{2\eta\|w\|^{2}}\left(e^{rT}|h_{1}-h_{2}|^{\frac{\zeta}{5(2-\gamma)}}+e^{-\Lambda
T}\right)$ $\displaystyle\leq Ce^{2\eta\|w\|^{2}}|h_{1}-h_{2}|^{\gamma_{1}}.$

with $\gamma_{1}=\frac{\Lambda\zeta}{5(\Lambda+r)(2-\gamma)}$. Hence $\chi\in
C^{\gamma_{1}}_{2\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$. This completes
the proof of this proposition.

∎

We are now in a position to give the martingale approximation. For $T\geq 0$,
let

(4.14)
$\displaystyle\int_{0}^{T}\widetilde{\phi}(X_{t})dt=\int_{0}^{N}\widetilde{\phi}(X_{t})dt+\int_{N}^{T}\widetilde{\phi}(X_{t})dt=M_{N}+R_{N,T},$

where $N$ is the integer part of $T$,

$M_{N}=\chi(X_{N})-\chi(X_{0})+\int_{0}^{N}\widetilde{\phi}(X_{t})dt$

is the Dynkin martingale (formally) and

$R_{N,T}=-\chi(X_{N})+\chi(X_{0})+\int_{N}^{T}\widetilde{\phi}(X_{t})dt$

is the reminder term. Let $Z_{N}=M_{N}-M_{N-1}$ for $N\geq 1$ be the
associated martingale difference. In what follows, we will show that $M_{N}$
is indeed a martingale and $R_{N,T}$ is a negligible term that vanishes as
$T\rightarrow\infty$. This will reduce the proof of Theorem 4.1-4.2 to the
proof of the limit theorems for the associated martingale sequence $M_{N}$.
Let $M_{T}=\chi(X_{T})-\chi(X_{0})+\int_{0}^{T}\widetilde{\phi}(X_{t})dt$ for
$T\geq 0$.

######  Lemma 4.8.

$\\{M_{T}\\}_{T\geq 0}$ is a zero mean martingale w.r.t the filtration
$\\{\mathcal{F}_{T}\\}$.

###### Proof.

The martingale property follows from the Markov property of the homogenized
process $X_{t}$ as follows.

$\displaystyle\mathbf{E}[M_{T}|\mathcal{F}_{s}]=\mathbf{E}[\chi(X_{T})|\mathcal{F}_{s}]-\chi(X_{0})+\int_{0}^{s}\mathbf{E}[\widetilde{\phi}(X_{u})|\mathcal{F}_{s}]du+\int_{s}^{T}\mathbf{E}[\widetilde{\phi}(X_{u})|\mathcal{F}_{s}]du.$

Since $X_{u}$ is $\mathcal{F}_{s}$ measurable for $0\leq u\leq s$, it follows
that

$\int_{0}^{s}\mathbf{E}[\widetilde{\phi}(X_{u})|\mathcal{F}_{s}]du=\int_{0}^{s}\widetilde{\phi}(X_{u})du.$

Moreover, by the Markov property,

$\displaystyle\int_{s}^{T}\mathbf{E}[\widetilde{\phi}(X_{u})|\mathcal{F}_{s}]du=\int_{s}^{\infty}P_{u-s}\widetilde{\phi}(X_{s})du-\int_{T}^{\infty}P_{u-T}(P_{T-s}\widetilde{\phi})(X_{s})du=\chi(X_{s})-\mathbf{E}[\chi(X_{T})|\mathcal{F}_{s}].$

Hence $\mathbf{E}[M_{T}|\mathcal{F}_{s}]=M_{s}$.

It is zero mean since

$\displaystyle M_{T}$
$\displaystyle=\chi(X_{T})-\chi(X_{0})+\int_{0}^{T}\widetilde{\phi}(X_{t})dt$
$\displaystyle=\int_{T}^{\infty}P_{t-T}\widetilde{\phi}(X_{T})dt-\int_{0}^{\infty}P_{t}\widetilde{\phi}(X_{0})dt+\int_{0}^{T}\widetilde{\phi}(X_{t})dt$
$\displaystyle=\int_{T}^{\infty}\mathbf{E}_{(w,h)}\left[\widetilde{\phi}(X_{t})|\mathcal{F}_{T}\right]dt-\int_{0}^{\infty}P_{t}\widetilde{\phi}(w,h)dt+\int_{0}^{T}\widetilde{\phi}(X_{t})dt,$

which implies $\mathbf{E}_{(w,h)}M_{T}$ = 0. ∎

The following lemma gives estimates on the even order moments of the
martingale $M_{N}$ and its associated martingale difference.

######  Lemma 4.9 (Bounds on the martingale).

For integer $p\geq 1$, $\eta\in(0,2^{-p-1}\eta_{0}]$ and $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, one has

$\mathbf{E}_{(w,h)}|M_{T}|^{2^{p}}\leq
C(T^{2-2^{-p}}+1)e^{2^{p+1}\eta\|w\|^{2}},\quad\mathbf{E}_{(w,h)}|Z_{N}|^{2^{p}}\leq
Ce^{2^{p+1}\eta\|w\|^{2}},$

for $T\geq 0$ and $N\geq 1$. Also with a larger constant $C$,

$P_{t}\mathbf{E}_{(w,h)}|M_{T}|^{2p}\leq
C(T^{2-2^{-p}}+1)e^{2^{p+1}\eta\|w\|^{2}},\quad\forall t\geq 0.$

###### Proof.

By Proposition 4.7, we know that $\chi\in
C^{\gamma_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})$. Hence $\chi^{2^{p}}\in
C^{\gamma_{0}}_{2^{p+1}\eta,H}(H\times\mathbb{T}^{n})$. Besides, since
$\phi\in C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$,
$|\widetilde{\phi}|^{2^{p}}\in
C^{\gamma}_{2^{p+1}\eta,H}(H\times\mathbb{T}^{n})$. It follows from estimate
(6.1) that for $\eta\in(0,2^{-p-1}\eta_{0}]$, and any $t\geq 0$,

$\displaystyle\mathbf{E}_{(w,h)}|M_{T}|^{2^{p}}$ $\displaystyle\leq
C\left(\mathbf{E}_{(w,h)}|\chi(X_{T})|^{2^{p}}+|\chi(w,h)|^{2^{p}}+T^{1-2^{-p}}\int_{0}^{T}\mathbf{E}_{(w,h)}|\widetilde{\phi}(X_{t})|^{2^{p}}dt\right)$
$\displaystyle\leq
C\left(\mathbf{E}e^{2^{p+1}\eta\|w_{0,T,h}(w)\|^{2}}+e^{2^{p+1}\eta\|w\|^{2}}+T^{1-2^{-p}}\int_{0}^{T}\mathbf{E}e^{2^{p+1}\eta\|w_{0,t,h}(w)\|^{2}}dt\right)$
$\displaystyle\leq C(T^{2-2^{-p}}+1)e^{2^{p+1}\eta\|w\|^{2}}.$

Similarly, one can show that for any $N\geq 1$,

$\displaystyle\mathbf{E}_{(w,h)}|Z_{N}|^{2^{p}}\leq
Ce^{2^{p+1}\eta\|w\|^{2}},$

where $C$ does not depend on $N,h$. It follows from (6.1) that

$P_{t}\mathbf{E}_{(w,h)}|M_{T}|^{2^{p}}\leq
C(T^{2-2^{-p}}+1)\mathbf{E}e^{2^{p+1}\eta\|w_{0,t,h}(w)\|^{2}}\leq
C(T^{2-2^{-p}}+1)e^{2^{p+1}\eta\|w\|^{2}}.$

∎

The following estimate on the remainder term shows that the proof of Theorem
4.1-4.2 can be reduced to the proof of the corresponding limit theorems for
the associated martingale sequence.

######  Lemma 4.10.

Let $R_{N,T}$ be as in (4.14). Then for any initial condition $X_{0}=(w,h)$,
$\eta\in(0,2^{-4}\eta_{0}]$ and $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$,

(4.15)
$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{\sqrt{T}}R_{N,T}=0,\quad\mathbf{P}\text{-}\mathrm{a.s}...$

###### Proof.

Since $N$ is the integer part of $T$, it suffices to show

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{\sqrt{N}}\sup_{N\leq t\leq
N+1}R_{N,t}=0,\quad\mathbf{P}\text{-}\mathrm{a.s}...$

By Proposition 4.7, we have that

$|\chi(X_{N})|\leq Ce^{2\eta\|w_{0,N,h}(w)\|^{2}}.$

Since $\phi\in C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})$, it also holds that

(4.16) $\displaystyle\sup_{N\leq t\leq
N+1}\left|\int_{N}^{t}\widetilde{\phi}(X_{s})ds\right|\leq C\sup_{N\leq t\leq
N+1}e^{2\eta\|w_{0,t,h}(w)\|^{2}}.$

It then follows from the Markov inequality, estimates (6.2) and (6.1) that for
any $K>0$,

$\displaystyle\mathbf{P}\left(\sup_{N\leq t\leq
N+1}e^{2\eta\|w_{0,t,h}(w)\|^{2}}>K\right)\leq Ce^{2^{4}\eta\|w\|^{2}}K^{-8}.$

Hence

$\displaystyle\sum_{N=1}^{\infty}\mathbf{P}\left(\sup_{N\leq t\leq
N+1}\left(|\chi(X_{N})|+\chi(w,h)+\left|\int_{N}^{t}\widetilde{\phi}(X_{s})ds\right|\right)\geq
N^{\frac{1}{4}}\right)$
$\displaystyle\leq\sum_{N=1}^{\infty}\mathbf{P}\left(C\sup_{N\leq t\leq
N+1}e^{2\eta\|w_{0,t,h}(w)\|^{2}}\geq N^{\frac{1}{4}}\right)\leq
Ce^{2^{4}\eta\|w\|^{2}}\sum_{N=1}^{\infty}N^{-2}<\infty,$

By the Borel-Cantelli lemma, there is an almost surely finite random integer
time $N_{0}(\omega)$ such that for $N>N_{0}(\omega)$,

(4.17) $\displaystyle\sup_{N\leq t\leq N+1}R_{N,t}\leq N^{1/4},$

which implies (4.15). ∎

###  4.2. The limit theorems

Based on the martingale approximation given above, we will prove Theorem 4.1
and Theorem 4.2 in this subsection by showing the limit results for the
corresponding martingales. As remarked earlier, compared with the analysis in
the next subsection on the rate of convergence, the proof of the limit
theorems is quite straightforward that merely requires certain moment bounds
as given in Lemma 4.9 and ergodicity properties given in Proposition 3.11. We
first prove the strong law of large numbers, which is based on the following
Kolmogorov’s criterion for martingales.

######  Theorem 4.11 ([22, 28]).

Let $\\{M_{N}\\}_{N\geq 1}$ be a zero mean square integrable martingale and
let $\\{c_{N}\\}$ be an increasing sequence going to $\infty$ such that

$\sum_{N=1}^{\infty}c_{N}^{-2}\mathbf{E}Z_{N}^{2}<\infty,$

where $Z_{N}=M_{N}-M_{N-1}$ and $M_{0}=0$. Then

$\lim_{N\rightarrow\infty}c_{N}^{-1}M_{N}\rightarrow
0,\quad\mathbf{P}\text{-}\mathrm{a.s}..$

###### Proof of Theorem 4.1.

In view of the martingale approximation (4.14) and Theorem 4.10, to show
(4.6), it suffices to prove that

$\displaystyle\lim_{N\rightarrow\infty}N^{-\frac{1}{2}-\varepsilon}M_{N}=0,\quad\mathbf{P}\text{-}\mathrm{a.s}..$

Lemma 4.9 with $p=1$ ensures the condition of Theorem 4.11 with
$c_{N}=N^{1/2+\varepsilon}$. Hence the desired convergence follows from
Theorem 4.11. ∎

The rest of this subsection is devoted to the proof of Theorem 4.2. We first
prove the existence of the asymptotic variance.

######  Proposition 4.12 (The asymptotic variance).

For any $(w,h)\in H\times\mathbb{T}^{n}$, $\eta\in(0,\eta_{0}/16]$ and
$\phi\in C_{\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$, we have

(4.18)
$\displaystyle\lim_{T\rightarrow\infty}\frac{1}{T}\mathbf{E}\left(\int_{0}^{T}\widetilde{\phi}\left(X_{t}(w,h)\right)dt\right)^{2}=2\int_{H\times\mathbb{T}^{n}}\widetilde{\phi}(w,h)\chi(w,h)\Gamma_{h}(dw)\lambda(dh):=\sigma_{\phi}^{2}.$

###### Proof.

By the Markov property of the homogenized process, one has

$\displaystyle\frac{1}{T}\mathbf{E}\left(\int_{0}^{T}\widetilde{\phi}\left(X_{t}(w,h)\right)dt\right)^{2}$
$\displaystyle=\frac{2}{T}\mathbf{E}\int_{0}^{T}\int_{s}^{T}\widetilde{\phi}\left(X_{t}(w,h)\right)\widetilde{\phi}\left(X_{s}(w,h)\right)dtds$
$\displaystyle=\frac{2}{T}\int_{0}^{T}\mathbf{E}\left[\widetilde{\phi}\left(X_{s}(w,h)\right)\int_{s}^{T}\mathbf{E}\left[\widetilde{\phi}\left(X_{t}(w,h)\right)|\mathcal{F}_{s}\right]dt\right]ds$
$\displaystyle=\frac{2}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\widetilde{\phi}\int_{0}^{T-s}P_{t}\widetilde{\phi}dt\right\rangle
ds.$

In view of the weak convergence in Proposition 3.11 and the definition of the
corrector in Proposition 4.7, we expect that as $T\rightarrow\infty$,

$\displaystyle\Big{|}\frac{2}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\widetilde{\phi}\int_{0}^{T-s}P_{t}\widetilde{\phi}dt\right\rangle
ds$ $\displaystyle-\frac{2}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\widetilde{\phi}\chi\right\rangle ds\Big{|}$ (4.19)
$\displaystyle=\left|\frac{2}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\widetilde{\phi}(\chi-\chi_{T-s})\right\rangle
ds\right|\rightarrow 0,$

where $\chi_{T-s}=\int_{0}^{T-s}P_{t}\widetilde{\phi}dt$. Indeed, it follows
from Theorem 4.6 with $\kappa=2$ that

$|\chi-\chi_{T-s}|\leq\int_{T-s}^{\infty}\left|P_{t}\widetilde{\phi}(w,h)\right|dt\leq
Ce^{2\eta\|w\|^{2}}e^{-\Lambda(T-s)}.$

Since $\widetilde{\phi}\in C_{\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$, and
$e^{2\eta\|\cdot\|^{2}}\in C_{4\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$, it
follows that $\widetilde{\phi}e^{2\eta\|\cdot\|^{2}}\in
C_{8\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$. Hence by estimate (6.1), we have
for $\eta\in(0,\eta_{0}/8]$, and any $s\geq 0$,

$\displaystyle\left\langle
P_{s}^{*}\delta_{(w,h)},|\widetilde{\phi}|e^{2\eta\|\cdot\|^{2}}\right\rangle\leq
C\left\langle
P_{s}^{*}\delta_{(w,h)},e^{8\eta\|\cdot\|^{2}}\right\rangle=C\mathbf{E}e^{8\eta\|w_{0,s,h}(w)\|^{2}}\leq
Ce^{8\eta\|w\|^{2}}.$

Hence

$\displaystyle\left|\frac{1}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\widetilde{\phi}(\chi-\chi_{T-s})\right\rangle
ds\right|\leq
Ce^{8\eta\|w\|^{2}}\frac{1}{T}\int_{0}^{T}e^{-\Lambda(T-s)}ds\rightarrow 0,$

which implies the limit (4.19).

By Proposition 4.7, $\chi\in C^{\gamma_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})$.
Hence $|\widetilde{\phi}\chi|^{2}\in
C^{\gamma_{0}}_{8\eta,H}(H\times\mathbb{T}^{n})$. Then by estimate (6.1), for
$\eta\in(0,\eta_{0}/8]$,

$\displaystyle\limsup_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\left|\widetilde{\phi}\chi\right|^{2}\right\rangle
ds\leq
C\limsup_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\mathbf{E}e^{8\eta\|w_{0,s,h}(w)\|^{2}}ds<\infty.$

Combining this moment bound with the weak convergence in Proposition 3.11, we
obtain the desired convergence

$\displaystyle\lim_{T\rightarrow\infty}\frac{2}{T}\int_{0}^{T}\left\langle
P_{s}^{*}\delta_{(w,h)},\widetilde{\phi}\chi\right\rangle
ds=2\int_{H\times\mathbb{T}^{n}}\widetilde{\phi}(w,h)\chi(w,h)\Gamma_{h}(dw)\lambda(dh),$

which combined with (4.19) implies the desired (4.18). ∎

The following proposition gives further properties related to the asymptotic
variance. In particular, the Hölder regularity of for the particular
observable function $F$ as below, plays an important role when estimating the
rate of convergence in the central limit theorem.

######  Proposition 4.13.

For $\gamma\in(0,1]$, $\eta\in(0,2^{-5}\eta_{0}]$, $\phi\in
C^{\gamma}_{\eta,H}(H\times\mathbb{T}^{n})\cap
C_{\eta,\mathbb{T}^{n}}^{\gamma}(H\times\mathbb{T}^{n})$, and $X_{0}=(w,h)$,
let

$Y(w,h)=\mathbf{E}_{(w,h)}M_{1}^{2}=\mathbf{E}_{(w,h)}\left(\chi(X_{1})-\chi(X_{0})+\int_{0}^{1}\widetilde{\phi}(X_{t})dt\right)^{2}.$

Assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$. Then $Y\in
C^{\overline{\gamma}_{0}}_{2^{5}\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{2^{5}\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$.
Furthermore, the function

$F(h):=\int_{H}Y(w,h)\Gamma_{h}(dw)=\langle\Gamma_{h},Y(\cdot,h)\rangle$

is in $C^{\overline{\gamma}_{0}}(\mathbb{T}^{n},\mathbb{R})$. Here
$\overline{\gamma}_{0}$ is taken from Proposition 4.7. We also have

(4.20)
$\displaystyle\sigma^{2}=\int_{\mathbb{T}^{n}}F(h)\lambda(dh)=\int_{H\times\mathbb{T}^{n}}Y(w,h)\Gamma_{h}(dw)\lambda(dh).$

###### Proof.

It follows from the Markov property that

(4.21) $\displaystyle Y(w,h)=\chi^{2}(w,h)$
$\displaystyle+P_{1}\chi^{2}(w,h)-2\chi(w,h)P_{1}\chi(w,h)+2\int_{0}^{1}P_{t}(\widetilde{\phi}P_{1-t}\chi)(w,h)dt$
$\displaystyle-2\chi(w,h)\int_{0}^{1}P_{t}\widetilde{\phi}(w,h)dt+2\int_{0}^{1}\int_{0}^{t}P_{\tau}\left(\widetilde{\phi}P_{t-\tau}\widetilde{\phi}\right)(w,h)d\tau
dt.$

By Proposition 4.7 we know that $\chi\in
C^{\overline{\gamma}_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})\cap