is straightforward to check that the Markov operators $T^{V}_{0,t,h}$ satisfy
the cocycle property as well,

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

In addition, one directly verifies that the induced operator
$\mathcal{T}^{V}_{t}$ acting on $C_{b}(H\times\mathbb{T})$ by

(4.22)
$\displaystyle\mathcal{T}^{V}_{t}\phi(w,h)=T^{V}_{0,t,h}\phi(\cdot,\beta_{t}h)(w),\,\phi\in
C_{b}(H\times\mathbb{T}),$

is a Markov semigroup, whose restriction on $C_{b}(H)$ yields $T^{V}_{0,t,h}$.

Note that the action of $\mathcal{T}^{V}_{t}$ on $C_{b}(\mathbb{T})$ is just
the Koopman operator for the circle rotation. Hence any invariant measure
$\mu$ of $\mathcal{T}^{V}_{t}$ must have marginal $m$ on $\mathbb{T}$. By
disintegration theorem, the disintegration $\mu(h),h\in\mathbb{T}$ of $\mu$ is
uniquely determined up to an $m$-null set and $\mu=\mu(h)m(dh)$. One can check
that $\mu(h)$ is an invariant measure of the evolution family
$T_{0,t,h}^{V*}$, where $T_{0,t,h}^{V*}$ is the dual of $T_{0,t,h}^{V}$.
Namely, for every $t\geq 0$,

(4.23) $\displaystyle T_{0,t,h}^{V*}\mu(h)=\mu(\beta_{t}h),\quad m\text{-a.e
}h\in\mathbb{T}.$

Conversely, any measurable family of probability measures
$\mu(h),h\in\mathbb{T}$ satisfying (4.23) gives an invariant measure
$\mu(h)m(dh)$ of $\mathcal{T}^{V}_{t}$. In this vein, unique ergodicity of
Markov evolution family (4.20) is equivalent to the unique ergodicity of its
homogenization (4.22) and hence we will not distinguish them.

The following lemma says that the semigroup $\mathcal{T}^{V}_{t}$ is uniquely
ergodic with unique invariant measure
$F^{V}(h)\Gamma^{V}(h)m(dh)\in\mathcal{P}(H\times\mathbb{T})$, which is a
consequence of the uniqueness of eigenmeasure from Theorem 2.1.

######  Lemma 4.2.

Let $\gamma_{V}(h)=F^{V}(\cdot,h)\Gamma^{V}(h)$. Then $\gamma_{V}(h)m(dh)$ is
the unique invariant measure of the semigroup $\mathcal{T}^{V}_{t}$.

###### Proof.

We first verify that $\gamma_{V}(h)m(dh)$ is indeed invariant under
$\mathcal{T}^{V}_{t}$. Since $\Gamma^{V}$ is the eigenmeasure of
$P_{0,t,h}^{V}$, one has

$\displaystyle\left\langle T_{0,t,h}^{V}g,\gamma_{V}(h)\right\rangle$
$\displaystyle=\left\langle\lambda_{t,h,V}^{-1}F^{V}(\cdot,h)^{-1}(P_{0,t,h}^{V}g(\cdot)F^{V}(\cdot,\beta_{t}h)),F^{V}(\cdot,h)\Gamma^{V}(h)\right\rangle$
$\displaystyle=\left\langle\lambda_{t,h,V}^{-1}g(\cdot)F^{V}(\cdot,\beta_{t}h),P_{0,t,h}^{V*}\Gamma^{V}(h)\right\rangle=\left\langle
g,F^{V}(\cdot,\beta_{t}h)\Gamma^{V}(\beta_{t}h)\right\rangle,$

for every $g\in C_{b}(H)$. Hence $\gamma_{V}$ is an invariant measure for
$T_{0,t,h}^{V*}$, i.e.,

$T_{0,t,h}^{V*}\gamma_{V}(h)=\gamma_{V}(\beta_{t}h).$

Therefore $\gamma_{V}(h)m(dh)$ is an invariant measure for the semigroup
$\mathcal{T}^{V}_{t}$.

We next show that this invariant measure is unique. Suppose that
$\mu\in\mathcal{P}(H\times\mathbb{T})$ is a different invariant measure of the
semigroup $\mathcal{T}^{V}_{t}$, then it must have marginal $m$ on
$\mathbb{T}$ and its unique disintegration which we denote by $\gamma(h)$
satisfies (4.23), i,e.,

$T_{0,t,h}^{V*}\gamma(h)=\gamma(\beta_{t}h),\quad m\text{-a.e
}h\in\mathbb{T}.$

We claim that $\gamma$ is necessarily equal to $\gamma_{V}$ up to a null set.
Let $\gamma_{0}(h)=F^{V}(\cdot,h)^{-1}\gamma(h)$ and $\phi\in C_{b}(H)$. It
follows from the invariance of $\gamma$ that

$\displaystyle\left\langle\gamma_{0}(\beta_{t}h),\phi
F^{V}(\cdot,\beta_{t}h)\right\rangle=\left\langle\gamma(\beta_{t}h),\phi\right\rangle$
$\displaystyle=\left\langle T_{0,t,h}^{V*}\gamma(h),\phi\right\rangle$
$\displaystyle=\lambda_{t,h,V}^{-1}\left\langle
F^{V}(\cdot,h)^{-1}\gamma(h),P_{0,t,h}^{V}\left(\phi(\cdot)F^{V}(\cdot,\beta_{t}h)\right)\right\rangle$
$\displaystyle=\lambda_{t,h,V}^{-1}\left\langle
P_{0,t,h}^{V*}\gamma_{0}(h),\phi(\cdot)F^{V}(\cdot,\beta_{t}h)\right\rangle.$

Since $\phi$ is arbitrary and $F^{V}$ is strictly positive, we deduce

$P_{0,t,h}^{V*}\gamma_{0}(h)=\lambda_{t,h,V}\gamma_{0}(\beta_{t}h),\quad
m\text{-a.e }h\in\mathbb{T}.$

The uniqueness of eigenmeasure then implies $\gamma_{0}=\Gamma^{V}$ up to a
null set, which in turn shows that $\gamma=\gamma_{V}$ for almost every
$h\in\mathbb{T}$. Hence $\gamma_{V}(h)m(dh)$ is the unique invariant measure
of the semigroup $T_{t}^{V}$.

The proof of Lemma 4.2 is complete. ∎

Step 2: Correspondence between equilibrium states and zeros of the auxiliary
rate function obtained from the auxiliary Markov evolution operators (4.20).
To show the uniqueness of equilibrium state associated with $V\in
C_{b}^{1,0}(H\times\mathbb{T})$ as in (4.4), we shall establish a
correspondence between equilibrium states and the zeros of an auxiliary rate
function, where the later is connected with the invariant measure of
$\mathcal{T}_{t}^{V}$. The uniqueness of the invariant measure proved in Lemma
4.2 thus will imply the uniqueness of equilibrium state.

More precisely, consider the auxiliary Feynman-Kac operators $R_{0,t,h}^{V,U}$
associated with $T_{0,t,h}^{V}$ and $U\in C_{b}(H\times\mathbb{T})$ given by

$R_{0,t,h}^{V,U}g(w)=\lambda_{t,h,V}^{-1}F^{V}(w,h)^{-1}(P_{0,t,h}^{V+U}g(\cdot)F^{V}(\cdot,\beta_{t}h))(w),\quad
g\in C_{b}(H),(w,h)\in H\times\mathbb{T},$

which induces as in (4.22) its homogenization $\mathcal{R}_{t}^{V,U}$ acting
on $C_{b}(H\times\mathbb{T})$:

(4.24)
$\displaystyle\mathcal{R}_{t}^{V,U}g(w,h)=\lambda_{t,h,V}^{-1}F^{V}(w,h)^{-1}(P_{0,t,h}^{V+U}g(\cdot,\beta_{t}h)F^{V}(\cdot,\beta_{t}h))(w),\quad
g\in C_{b}(H\times\mathbb{T}).$

We claim that the following limit is well defined, which gives the
corresponding auxiliary pressure function,

(4.25) $\displaystyle
Q^{V}(U):=\lim_{t\to\infty}\frac{1}{t}\log(R_{0,t,h}^{V,U}\mathbb{1}_{H})(w),\quad
U\in C_{b}(H\times\mathbb{T}),w\in H,h\in\mathbb{T},$

which is independent of $w,h$.

Indeed, for $U\in C_{b}^{1,0}(H\times\mathbb{T})$, the forward convergence
(2.11) in Theorem 2.1 implies that

(4.26)
$\displaystyle\lim_{t\to\infty}\left|\lambda_{t,h,V+U}^{-1}P_{0,t,h}^{V+U}F^{V}(\cdot,\beta_{t}h)(w)-F^{V+U}(w,h)\left\langle\Gamma^{V+U}(\beta_{t}h),F^{V}(\cdot,\beta_{t}h)\right\rangle\right|=0.$

Estimating as in (3.51) and using the properties of eigenfunction $F^{V}$ and
eigenmeasure $\Gamma^{V+U}$ as in Proposition 3.2 and Proposition 3.1, one can
show that

$\displaystyle
p_{1}\leq\liminf_{t\to\infty}\left\langle\Gamma^{V+U}(\beta_{t}h),F^{V}(\cdot,\beta_{t}h)\right\rangle\leq\limsup_{t\to\infty}\left\langle\Gamma^{V+U}(\beta_{t}h),F^{V}(\cdot,\beta_{t}h)\right\rangle\leq
p_{2},$

for some constants $p_{1},p_{2}>0$. Therefore convergence (4.26) implies

$\displaystyle\lim_{t\to\infty}\left|\lambda_{t,h,V+U}^{-1}\left\langle\Gamma^{V+U}(\beta_{t}h),F^{V}(\cdot,\beta_{t}h)\right\rangle^{-1}P_{0,t,h}^{V+U}F^{V}(\cdot,\beta_{t}h)(w)-F^{V+U}(w,h)\right|=0.$

Combining this with Birkhoff’s ergodic theorem and continuity of
$h\to\lambda^{V+U}(h)$ and integral representation (2.10) for potential $V+U$,
we have

(4.27) $\displaystyle\begin{split}\lim_{t\to\infty}\frac{1}{t}\log
P_{0,t,h}^{V+U}F^{V}(\cdot,\beta_{t}h)(w)&=\lim_{t\to\infty}\frac{1}{t}\log\lambda_{t,h,V+U}\\\
&=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\lambda^{V+U}(\beta_{r}h)dr=\int_{\mathbb{T}}\lambda^{V+U}(h)m(dh)=Q(U+V).\end{split}$

In view of (4.25) and the definition of $R_{0,t,h}^{V,U}$, limits (4.19) and
(4.27) yield

(4.28) $\displaystyle Q^{V}(U)=Q(U+V)-Q(V),\,\text{ for }U\in
C_{b}^{1,0}(H\times\mathbb{T}).$

Therefore the limit (4.25) is well defined for $U\in
C_{b}^{1,0}(H\times\mathbb{T})$. The existence of the limit (4.25) for general
$U\in C_{b}(H\times\mathbb{T})$ follows from a $buc$-approximation argument as
in the proof of Item (1). The relation (4.28) extends to $U\in
C_{b}(H\times\mathbb{T})$ as well by Lipschitz continuity of the pressure
functions.

Denote by $I^{V}(\mu):=\displaystyle\sup_{U\in
C_{b}(H\times\mathbb{T})}\big{(}\left\langle
U,\mu\right\rangle-Q^{V}(U)\big{)}$ the Legendre transformation of $Q^{V}$,
which is the auxiliary rate function. Then relation (4.28) implies

$I^{V}(\mu)=I(\mu)+Q(V)-\langle V,\mu\rangle.$

Hence a measure $\mu_{V}$ is an equilibrium state of the potential $V$, i.e,

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

if and only if $I^{V}(\mu_{V})=0$.

Step 3: The zeros of the auxiliary rate function are invariant measures of the
Markov evolution operators (4.20). Following the scheme as in [29], one can
prove the following lemma that connects the zeros of the auxiliary rate
function $I^{V}$ with the invariant measure of $\mathcal{T}_{t}^{V}$. The
major difference is in proving the time continuity and Feller property of the
semigroup $\mathcal{T}_{t}^{V}$ since we are working with a semigroup obtained
by coupling the evolution operator (4.20) with the circle rotation. The proof
is postponed in the next subsection 4.2 for the reader’s convenience.

######  Lemma 4.3.

If $I^{V}(\mu)=0$ then $\mu$ is an invariant measure of the semigroup
$\mathcal{T}_{t}^{V}$.

Now since $I$ is a good rate function, there is at least one equilibrium state
for $V$. Hence the zeros of $I^{V}$ is non-empty. In view of Lemma 4.2 and
Lemma 4.3, we know that the equilibrium state for $V$ is unique, which is the
unique invariant measure $F^{V}(h)\Gamma^{V}(h)m(dh)$ of
$\mathcal{T}_{t}^{V}$. This proves Item (3) in Theorem 4.1.

The proof of Theorem 4.1 is complete. ∎

###  4.2. Proof of auxiliary lemmas

We give a proof of Lemma 4.3 in this subsection. We first establish time
continuity and Feller property of the semigroup $\mathcal{T}_{t}^{V}$ in
subsection 4.2.1, which is the part that is different from the time
homogeneous setting. Then in subsection 4.2.2 we prove Lemma 4.3, by combining
the time continuity and Feller property, following the scheme as in [29].
Recall that $V\in C_{b}^{1,0}(H\times\mathbb{T})$ and the semigroup
$\mathcal{T}_{t}^{V}$ is defined as in (4.22) through Doob’s $h$-transform
(4.20).

####  4.2.1. Time continuity and Feller property of $\mathcal{T}_{t}^{V}$

The following lemma establishes the time continuity and Feller property of the
semigroup.

######  Lemma 4.4.

The semigroup $\mathcal{T}_{t}^{V}$ has the following two properties:

  1. (1)

For any $\phi\in C_{b}(H\times\mathbb{T})$, and $(w,h)\in H\times\mathbb{T}$,
the function $t\to\mathcal{T}_{t}^{V}\phi(w,h)$ is continuous.

  2. (2)

For any $t\geq 0$, $\mathcal{T}_{t}^{V}$ maps $C_{b}(H\times\mathbb{T})$ to
itself.

###### Proof.

The proof is divided into the following two steps.

Step 1: Proof of time continuity. It follows from the integral representation
(2.10) for the eigenvalue that $\lambda_{t,h,V}$ is continuous in time. In
view of definition (4.20) and (4.22), we only need to show the time continuity
of the function

(4.29) $\displaystyle t\in[0,\infty)\to
P^{V}_{0,t,h}(\phi(\cdot,\beta_{t}h)F^{V}(\cdot,\beta_{t}h))(w).$

Let’s write for convenience $F_{\phi}=\phi F^{V}$. Since $\phi\in
C_{b}(H\times\mathbb{T})$, and $F^{V}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$, we know $F_{\phi}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$ as well. In addition, since
$F^{V}\in C(H\times\mathbb{T})$ by Theorem 2.1, we know that

$(w,\tau)\to F^{V}(w,\beta_{\tau}h)\,\text{ is jointly continuous
everywhere}.$

Therefore so is $F_{\phi}$.

By triangle inequality, denoting
$\Xi_{t}^{V}=\exp\left(\int_{0}^{t}V(\mathcal{H}_{r})dr\right)$,

(4.30)
$\displaystyle\begin{split}&\Big{|}P^{V}_{0,t,h}(\phi(\cdot,\beta_{t}h)F^{V}(\cdot,\beta_{t}h))(w)-P^{V}_{0,\tau,h}(\phi(\cdot,\beta_{\tau}h)F^{V}(\cdot,\beta_{\tau}h))(w)\Big{|}\\\
&\leq\mathbf{E}_{(w,h)}\Big{|}\Xi_{t}^{V}F_{\phi}(\mathcal{H}_{t})-\Xi_{\tau}^{V}F_{\phi}(\mathcal{H}_{t})\Big{|}+\mathbf{E}_{(w,h)}\Big{|}\Xi_{\tau}^{V}F_{\phi}(\mathcal{H}_{t})-\Xi_{\tau}^{V}F_{\phi}(\mathcal{H}_{\tau})\Big{|}\\\
&:=J_{1}+J_{2}.\end{split}$

Since $F_{\phi}\in L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$, we
obtain

(4.31) $\displaystyle\begin{split}J_{1}&\leq
C\mathbf{E}_{(w,h)}\left[\Big{|}\Xi_{t}^{V}-\Xi_{\tau}^{V}\Big{|}\mathfrak{m}_{q}(\Phi_{0,t,h})\right]\\\
&\leq
C\mathbf{E}_{(w,h)}\left[\Big{|}\int_{\tau}^{t}V(\mathcal{H}_{r})dr\Big{|}\mathfrak{m}_{q}(\Phi_{0,t,h})\right]\\\
&\leq C|t-\tau|\mathfrak{m}_{q}(w),\end{split}$

where in the second step we used the fact $V\in
C_{b}^{1,0}(H\times\mathbb{T})$, and the Lipschitz continuity of $e^{x}$ on
some bounded interval determined by $m_{V},M_{V},t,\tau$ and in the last
inequality we used (B.6). Hence $J_{1}\to 0$ as $t\to\tau$.

On the other hand,

$\displaystyle J_{2}\leq
e^{\tau\|V\|_{\infty}}\mathbf{E}_{w}\Big{|}F_{\phi}(\Phi_{0,t,h},\beta_{t}h)-F_{\phi}(\Phi_{0,\tau,h},\beta_{\tau}h)\Big{|}$

By time continuity of sample paths of the solution $\Phi_{0,t,h}(w)$ and joint
continuity of $(w,t)\to F_{\phi}(w,\beta_{t}h)$, we know that

$\displaystyle|F_{\phi}(\Phi_{0,t,h},\beta_{t}h)-F_{\phi}(\Phi_{0,\tau,h},\beta_{\tau}h)\Big{|}\to
0$

almost surely as $t\to\tau$. In addition, the fact $F_{\phi}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$ implies

$\displaystyle\mathbf{E}_{w}\Big{|}F_{\phi}(\Phi_{0,t,h},\beta_{t}h)-F_{\phi}(\Phi_{0,\tau,h},\beta_{\tau}h)\Big{|}\leq
C\mathfrak{m}_{q}(w),$

by (B.6). Hence by Lebesgue’s dominated convergence theorem, we have $J_{2}\to
0$ as $t\to\tau$.

In view of (4.30), we obtain the desired continuity of (4.29). The proof of
Step 1 is complete.

Step 2: Proof of Feller property. Fix $t\geq 0$. We first show that for any
Lipschitz $\phi\in C_{b}(H\times\mathbb{T})$,

(4.32) $\displaystyle(w,h)\in
H\times\mathbb{T}\to\mathcal{T}_{t}^{V}\phi(w,h)\,\text{ is continuous. }$

Since $(w,h)\to\frac{1}{\lambda_{t,h,V}F^{V}(w,h)}$ is continuous by Theorem
2.1 , in view of definitions (4.20) and (4.22), we only need to show the
continuity of the function

(4.33) $\displaystyle(w,h)\in
H\times\mathbb{T}\to\mathcal{P}_{t}^{V}F_{\phi}(w,h),$

where $F_{\phi}=\phi F^{V}$. The main difficulties here are possible
unboundedness of $F^{V}$ and coupling of the Navier-Stokes family (2.2) with
the circle rotation.

Denoting
$\Xi_{t}^{V}(w,h)=\exp\left(\int_{0}^{t}V(\mathcal{H}_{r}(w,h))dr\right)$, we
write

(4.34)
$\displaystyle\begin{split}\mathcal{P}_{t}^{V}F_{\phi}(w_{n},h_{n})-\mathcal{P}_{t}^{V}F_{\phi}(w,h)&=\mathbf{E}_{(w_{n},h_{n})}\Xi_{t}^{V}F_{\phi}(\mathcal{H}_{t})-\mathbf{E}_{(w,h)}\Xi_{t}^{V}F_{\phi}(\mathcal{H}_{t})\\\
&=\mathbf{E}\left[\left(\Xi_{t}^{V}(w_{n},h_{n})-\Xi_{t}^{V}(w,h)\right)F_{\phi}(\mathcal{H}_{t}(w_{n},h_{n}))\right]\\\
&\qquad+\mathbf{E}\left[\Xi_{t}^{V}(w,h)\big{(}F_{\phi}(\mathcal{H}_{t}(w_{n},h_{n}))-F_{\phi}(\mathcal{H}_{t}(w,h))\big{)}\right]\\\
&:=I_{1}+I_{2}.\end{split}$

We can assume $w_{n},w\in B_{R_{0}}^{H}$ for some $R_{0}>0$ and all $n\geq 1$.
Note that since $V\in C_{b}^{1,0}(H\times\mathbb{T})$, by the same reasoning
as in (4.31), and the fact $F_{\phi}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$ as well as (B.6), we have

(4.35) $\displaystyle\begin{split}|I_{1}|&\leq
C\mathbf{E}\left[\int_{0}^{t}\left|V(\mathcal{H}_{r}(w_{n},h_{n}))-V(\mathcal{H}_{r}(w,h))\right|dr\mathfrak{m}_{q}(\Phi_{0,t,h_{n}}(w_{n}))\right]\\\
&\leq
C\left(\mathbf{E}\int_{0}^{t}\left|V(\mathcal{H}_{r}(w_{n},h_{n}))-V(\mathcal{H}_{r}(w,h))\right|^{2}dr\right)^{\frac{1}{2}}\left(\mathbf{E}\mathfrak{m}_{2q}(\Phi_{0,t,h_{n}}(w_{n}))\right)^{\frac{1}{2}}\\\
&\leq
C\left(\mathbf{E}\int_{0}^{t}\left|V(\mathcal{H}_{r}(w_{n},h_{n}))-V(\mathcal{H}_{r}(w,h))\right|^{2}dr\right)^{\frac{1}{2}}.\end{split}$

By $V\in C_{b}^{1,0}(H\times\mathbb{T})$ and triangle inequality, we have the
existence of some modulus of continuity $\kappa$ such that

$\displaystyle\mathbf{E}\int_{0}^{t}\left|V(\mathcal{H}_{r}(w_{n},h_{n}))-V(\mathcal{H}_{r}(w,h))\right|^{2}dr$
$\displaystyle\leq
C\left(\int_{0}^{t}\kappa^{2}(|\beta_{r}h_{n}-\beta_{r}h|)dr+\mathbf{E}\int_{0}^{t}|\Phi_{0,r,h_{n}}(w_{n})-\Phi_{0,r,h}(w)|^{2}dr\right)$
$\displaystyle\leq
C\left(\kappa^{2}(|h_{n}-h|)+\mathbf{E}\int_{0}^{t}|\Phi_{0,r,h_{n}}(w_{n})-\Phi_{0,r,h_{n}}(w)|^{2}dr+\mathbf{E}\int_{0}^{t}|\Phi_{0,r,h_{n}}(w)-\Phi_{0,r,h}(w)|^{2}dr\right)$
$\displaystyle\leq
C\left(\kappa^{2}(|h_{n}-h|)+\|w_{n}-w\|^{2}+|h_{n}-h|^{2}\right),$

where we used (B.4) with Lipschitz continuity of $f$ and (B.5) in the last
inequity. Combining this with (4.35), we have

(4.36) $\displaystyle I_{1}\leq
C\left(\kappa(|h_{n}-h|)+\|w_{n}-w\|+|h_{n}-h|\right).$

We now estimate $I_{2}$. The boundedness of $V$ yields

(4.37)
$\displaystyle\begin{split}e^{-t\|V\|_{\infty}}|I_{2}|\leq&\mathbf{E}\left|F_{\phi}(\mathcal{H}_{t}(w_{n},h_{n}))-F_{\phi}(\mathcal{H}_{t}(w,h))\right|\\\
&\leq\mathbf{E}\left|F_{\phi}(\Phi_{0,t,h_{n}}(w_{n}),\beta_{t}h_{n})-F_{\phi}(\Phi_{0,t,h}(w),\beta_{t}h_{n})\right|\\\
&\qquad+\mathbf{E}\left|F_{\phi}(\Phi_{0,t,h}(w),\beta_{t}h_{n})-F_{\phi}(\Phi_{0,t,h}(w),\beta_{t}h)\right|\\\
&:=I_{21}+I_{22}.\end{split}$

Note that $\phi\in C_{b}(H\times\mathbb{T}^{d})$ is Lipschitz, and by
Proposition 3.8, for any $R>0$, the family $F^{V}(\cdot,h),h\in\mathbb{T}$ is
uniformly Lipschitz on $B_{R}:=B_{R}^{H}$ with a uniform bound $L=L(R)$ on the
Lipschitz constants, therefore so is the the family $F_{\phi}(\cdot,h)=(\phi
F^{V})(\cdot,h),h\in\mathbb{T}$. In addition, (B.4) with Lipschitz continuity
of $f$ and (B.5) imply that

(4.38)
$\displaystyle\begin{split}\mathbf{E}\|\Phi_{0,t,h_{n}}(w_{n})-\Phi_{0,t,h}(w)\|&\leq\mathbf{E}\|\Phi_{0,t,h_{n}}(w_{n})-\Phi_{0,t,h_{n}}(w)\|+\mathbf{E}\|\Phi_{0,t,h_{n}}(w)-\Phi_{0,t,h}(w)\|\\\
&\leq C\left(\|w_{n}-w\|+|h_{n}-h|\right),\end{split}$

with constant $C=C(R_{0})$, where we used the fact that $w_{n},w\in
B_{R_{0}}^{H}$ for all $n\geq 1$. Since
$F_{\phi}=\mathbb{1}_{B_{R}}F_{\phi}+\mathbb{1}_{B_{R}^{c}}F_{\phi}\in
L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$, we have

(4.39)
$\displaystyle\begin{split}I_{21}&\leq\mathbf{E}\left|(\mathbb{1}_{B_{R}}F_{\phi})(\Phi_{0,t,h_{n}}(w_{n}),\beta_{t}h_{n})-(\mathbb{1}_{B_{R}}F_{\phi})(\Phi_{0,t,h}(w),\beta_{t}h_{n})\right|\\\
&\qquad+\mathbf{E}\left|(\mathbb{1}_{B_{R}^{c}}F_{\phi})(\Phi_{0,t,h_{n}}(w_{n}),\beta_{t}h_{n})-(\mathbb{1}_{B_{R}^{c}}F_{\phi})(\Phi_{0,t,h}(w),\beta_{t}h_{n})\right|\\\
&\leq
L(R)\mathbf{E}\|\Phi_{0,t,h_{n}}(w_{n})-\Phi_{0,t,h}(w)\|+CP_{0,t,h_{n}}\left(\mathbb{1}_{B_{R}^{c}}\mathfrak{m}_{q}\right)(w_{n})+CP_{0,t,h}\left(\mathbb{1}_{B_{R}^{c}}\mathfrak{m}_{q}\right)(w).\end{split}$

By Hölder’s inequality and Markov inequality, as well as (B.6), we have

(4.40)
$\displaystyle\begin{split}P_{0,t,\mathfrak{h}}\left(\mathbb{1}_{B_{R}^{c}}\mathfrak{m}_{q}\right)(u)&\leq\left(P_{0,t,\mathfrak{h}}\mathbb{1}_{B_{R}^{c}}(u)\right)^{1/2}\left(P_{0,t,\mathfrak{h}}\mathfrak{m}_{2q}(u)\right)^{1/2}\\\
&\leq CR^{-1}P_{0,t,\mathfrak{h}}\|\cdot\|(u)\leq CR^{-1},\end{split}$

for all $(u,\mathfrak{h})\in\left\\{(w,h),(w_{n},h_{n}):n\geq 1\right\\}$,
where $C=C(R_{0})$. Hence by combining (4.38)-(4.40), we obtain

$\displaystyle I_{21}\leq C(R)\left(\|w_{n}-w\|+|h_{n}-h|\right)+CR^{-1},$

which tends to zero by first making $R$ large and then $n$ large so that
$\|w_{n}-w\|+|h_{n}-h|\to 0$.

We now look at $I_{22}$. Since for each $u\in H$, we have

$\displaystyle F_{\phi}(u,\beta_{t}h_{n})\to F_{\phi}(u,\beta_{t}h),\,\text{
as }h_{n}\to h.$

and $\left|F_{\phi}(u,\beta_{t}h_{n})-F_{\phi}(u,\beta_{t}h)\right|\leq
C\mathfrak{m}_{q}(u)$ by $L_{\mathfrak{m}_{q}}^{\infty}(H\times\mathbb{T})$,
which is $P_{0,t,h}(w,du)$ integrable by (B.6), hence by Lebesgue’s dominated
convergence theorem, we have

$\displaystyle
I_{22}=\int_{H}\left|F_{\phi}(u,\beta_{t}h_{n})-F_{\phi}(u,\beta_{t}h)\right|P_{0,t,h}(w,du)\to
0\,\text{ as }h_{n}\to h.$

In view of (4.37) we have $|I_{2}|\to 0$ as long as $(w_{n},h_{n})\to(w,h)$.

Combining this with (4.34) and (4.36), we see that continuity of (4.33) and
hence (4.32) is true for any Lipschitz $\phi\in C_{b}(H\times\mathbb{T})$.

Now for any $\phi\in C_{b}(H\times\mathbb{T})$, we can find a sequence
$\phi_{m}$ of bounded Lipschitz functions such that

$\phi_{M}:=\sup_{m}\|\phi_{m}\|_{\infty}<\infty,\text{ and }\sup_{(w,h)\in
K\times\mathbb{T}}|\phi_{m}(w,h)-\phi(w,h)|\to 0$

as $m\to\infty$ for any compact $K\subset H$. For any $(w_{0},h_{0})\in
H\times\mathbb{T}$ and $(w_{n},h_{n})\to(w_{0},h_{0})$ as $n\to\infty$, denote
$z_{n}=(w_{n},h_{n})$ for $n\geq 0$. By triangle inequality we have

(4.41)
$\displaystyle\begin{split}&\left|\mathcal{T}_{t}^{V}\phi(z_{n})-\mathcal{T}_{t}^{V}\phi(z_{0})\right|\\\
&\leq\left|\mathcal{T}_{t}^{V}\phi(z_{n})-\mathcal{T}_{t}^{V}\phi_{m}(z_{n})\right|+\left|\mathcal{T}_{t}^{V}\phi_{m}(z_{n})-\mathcal{T}_{t}^{V}\phi_{m}(z_{0})\right|+\left|\mathcal{T}_{t}^{V}\phi_{m}(z_{0})-\mathcal{T}_{t}^{V}\phi(z_{0})\right|.\end{split}$

We can assume the existence of $R_{0}>0$ such that $z_{n}\in
B_{R_{0}}^{H}\times\mathbb{T}$ for all $n$. For any
$z=(u,\mathfrak{h})\in\left\\{z_{n}:n\geq 0\right\\}$, and
$K_{R}:=B_{R}^{H_{2}}$ which is compact in $H$, we have

(4.42)
$\displaystyle\begin{split}&\left|\mathcal{T}_{t}^{V}\phi(z)-\mathcal{T}_{t}^{V}\phi_{m}(z)\right|\\\
&\leq\left|\mathcal{T}_{t}^{V}\left(\mathbb{1}_{K_{R}}\phi\right)(z)-\mathcal{T}_{t}^{V}\left(\mathbb{1}_{K_{R}}\phi_{m}\right)(z)\right|+\left|\mathcal{T}_{t}^{V}\left(\mathbb{1}_{K_{R}^{c}}\phi\right)(z)-\mathcal{T}_{t}^{V}\left(\mathbb{1}_{K_{R}^{c}}\phi_{m}\right)(z)\right|\\\
&\leq\sup_{(w,h)\in K_{R}\times
T}|\phi(w,h)-\phi_{m}(w,h)|+\frac{C(\|\phi\|_{\infty},\phi_{M},\|V\|_{\infty})}{\lambda_{t,\mathfrak{h},V}F^{V}(u,\mathfrak{h})}P_{0,t,\mathfrak{h}}(\mathbb{1}_{K_{R}^{c}}\mathfrak{m}_{q}(u)).\end{split}$

By Hölder’s inequality and Markov inequity as in (4.40), and using (B.6), we
have

(4.43) $\displaystyle
P_{0,t,\mathfrak{h}}(\mathbb{1}_{K_{R}^{c}}\mathfrak{m}_{q}(u))\leq CR^{-1},$

with constant $C=C(R_{0})$ independent of $u,\mathfrak{h},R$. Since
$h\to\lambda_{t,h,V}>0$ is continuous, $F^{V}$ is strictly positive
everywhere, and $z_{n}\to z_{0}$, we have

$\displaystyle\inf_{(u,\mathfrak{h})\in\left\\{z_{n}:n\geq
0\right\\}}\lambda_{t,\mathfrak{h},V}F^{V}(u,\mathfrak{h})>0.$

Combining this with (4.42) and (4.43), we have

$\displaystyle\left|\mathcal{T}_{t}^{V}\phi(z)-\mathcal{T}_{t}^{V}\phi_{m}(z)\right|\leq\sup_{(w,h)\in
K_{R}\times T}|\phi(w,h)-\phi_{m}(w,h)|+CR^{-1}$

for all $z\in\left\\{z_{n}:n\geq 0\right\\}$, where the constant $C$ is
independent of $z$ and $R$. In view of (4.41), for any $\varepsilon>0$, we can
first choose $R$ large and then take $m$ large so that

$\left|\mathcal{T}_{t}^{V}\phi(z_{n})-\mathcal{T}_{t}^{V}\phi(z_{0})\right|\leq\left|\mathcal{T}_{t}^{V}\phi_{m}(z_{n})-\mathcal{T}_{t}^{V}\phi_{m}(z_{0})\right|+\frac{\varepsilon}{2},$

for all $n$. Then using continuity of (4.32) for Lipschitz $\phi_{m}$, we find
$N=N(m,R,\varepsilon)$ such that for all $n\geq N$,

$\left|\mathcal{T}_{t}^{V}\phi(z_{n})-\mathcal{T}_{t}^{V}\phi(z_{0})\right|<\varepsilon,$

which completes the proof of Feller property.

The proof of Lemma 4.4 is complete.

∎

####  4.2.2. Proof of Lemma 4.3

In this subsection we prove Lemma 4.3 by following the scheme in [29].
Consider the infinitesimal generator $\mathbb{L}_{V}$ of the Markov semigroup
$\mathcal{T}_{t}^{V}$. The domain $\mathcal{D}(\mathbb{L}_{V})$ of the
generator consists of those functions $\phi\in C_{b}(H\times\mathbb{T})$ so
that there is some $\varphi\in C_{b}(H\times\mathbb{T})$ such that the
following is true,

(4.44)
$\displaystyle\mathcal{T}_{t}^{V}\phi(w,h)=\phi(w,h)+\int_{0}^{t}\mathcal{T}_{r}^{V}\varphi(w,h)dr,\,t\geq
0,(w,h)\in H\times\mathbb{T}.$

Therefore for every $\phi\in\mathcal{D}(\mathbb{L}_{V})$, it follows from the
time continuity in Lemma 4.4 that the associated function $\varphi$ in the
representation (4.44) is unique and it is equal to the following limit,

(4.45) $\displaystyle\varphi(w,h)=\lim_{t\to
0}\frac{\mathcal{T}_{t}^{V}\phi(w,h)-\phi(w,h)}{t},\,(w,h)\in
H\times\mathbb{T}.$

We then define $\varphi$ as the image of $\phi$ under the action of
$\mathbb{L}_{V}$ and write $\varphi=\mathbb{L}_{V}\phi$. Hence the integral
equation (4.44) can be rewritten as the differential equation

$\displaystyle\partial_{t}\mathcal{T}_{t}^{V}\phi=\mathcal{T}_{t}^{V}\mathbb{L}_{V}\phi,\,\phi\in\mathcal{D}(\mathbb{L}_{V}).$

It also follows from the definition (4.44) as well as Lemma 4.4 that if
$\phi\in\mathcal{D}(\mathbb{L}_{V})$ with $\varphi=\mathbb{L}_{V}\phi$ then
$\mathcal{T}_{s}^{V}\phi\in\mathcal{D}(\mathbb{L}_{V})$ with

$\mathbb{L}_{V}\mathcal{T}_{s}^{V}\phi=\mathcal{T}_{s}^{V}\varphi=\mathcal{T}_{s}^{V}\mathbb{L}_{V}\phi.$

The following Lemma 4.5 is useful. Its proof is given after we show the proof
of Lemma 4.3 below.

######  Lemma 4.5.

Recall the auxiliary pressure function $Q^{V}$ defined as in (4.25).

  1. (1)

The set

$\mathcal{D}_{+}:=\left\\{\phi\in\mathcal{D}(\mathbb{L}_{V}):\inf_{(w,h)\in
H\times\mathbb{T}}\phi(w,h)>0\right\\},$

is determining for $\mathcal{P}(H\times\mathbb{T})$, i.e., for any
$\mu_{1},\mu_{2}\in\mathcal{P}(H\times\mathbb{T})$, if
$\left\langle\phi,\mu_{1}\right\rangle=\left\langle\phi,\mu_{2}\right\rangle$
for all $\phi\in\mathcal{D}_{+}$ then $\mu_{1}=\mu_{2}$.

  2. (2)

For any $\phi\in\mathcal{D}_{+}$, we have $Q^{V}(U_{\phi})=0$ where
$U_{\phi}=-\frac{\mathbb{L}_{V}\phi}{\phi}$.

###### Poof of Lemma 4.3.

Assume $I^{V}(\mu)=0$. Since $Q^{V}(U_{\phi})=0$ and $U_{\phi}\in
C_{b}(H\times\mathbb{T})$, one has

$\displaystyle 0=I^{V}(\mu)=\sup_{U\in
C_{b}(H\times\mathbb{T})}\left(\left\langle
U,\mu\right\rangle-Q^{V}(U)\right)\geq\sup_{\phi\in\mathcal{D}_{+}}\left\langle
U_{\phi},\mu\right\rangle$

This is to say

$\displaystyle\inf_{\phi\in\mathcal{D}_{+}}\left\langle\frac{\mathbb{L}_{V}\phi}{\phi},\mu\right\rangle\geq
0.$

Since $\mathcal{T}_{t}^{V}$ is Markov, we have $\mathbb{L}_{V}\mathbb{1}=0$,
which implies that $\alpha=$ is a local minimum of the function

$\displaystyle
J_{\mu}(\alpha):=\left\langle\frac{\mathbb{L}_{V}\left(1+\alpha\phi\right)}{1+\alpha\phi},\mu\right\rangle,\,\text{
for any }\phi\in\mathcal{D}_{+}.$

Therefore there must holds

(4.46) $\displaystyle
0=J_{\mu}^{\prime}(0)=\left\langle\mathbb{L}_{V}\phi,\mu\right\rangle,$

for all $\phi\in\mathcal{D}_{+}$. Since
$\mathcal{T}_{s}^{V}\phi\in\mathcal{D}_{+}$ for any $\phi\in\mathcal{D}_{+}$
and $s\geq 0$, replacing $\phi$ in (4.46) with $\mathcal{T}_{s}^{V}\phi$ and
integrate from $0$ to $t$ we obtain

$\displaystyle\left\langle\mathcal{T}_{t}^{V}\phi,\mu\right\rangle=\left\langle\phi,\mu\right\rangle$

for all $t\geq 0$ and $\phi\in\mathcal{D}_{+}$. Since $\mathcal{D}_{+}$ is
determining by Lemma 4.5, we have $\mathcal{T}_{t}^{V*}\mu=\mu$, i.e., any
$\mu$ satisfying $I^{V}(\mu)=0$ is invariant under $\mathcal{T}_{t}^{V}$.

The proof of Lemma 4.3 is complete.

∎

###### Proof of Lemma 4.5.

The proof is divided into two steps.

Step 1: Proof of Item (1). Let
$\mu_{1},\mu_{2}\in\mathcal{P}(H\times\mathbb{T})$ and assume

(4.47)
$\displaystyle\left\langle\phi,\mu_{1}\right\rangle=\left\langle\phi,\mu_{2}\right\rangle,\,\text{
for all }\phi\in\mathcal{D}_{+}.$

It is known that the following set

$\displaystyle C_{b}^{+}(H\times\mathbb{T}):=\left\\{U\in
C_{b}(H\times\mathbb{T}):\inf_{(w,h)\in H\times\mathbb{T}}U(w,h)>0\right\\}$

is determining. We will show that $\mathcal{D}_{+}$ is rich enough to
approximate any function in $C_{b}^{+}(H\times\mathbb{T})$.

Take any $U\in C_{b}^{+}(H\times\mathbb{T})$. We claim that
$\overline{U}_{s}:=\frac{1}{s}\int_{0}^{s}\mathcal{T}_{r}^{V}Udr\in\mathcal{D}_{+}$
for any $s>0$. Indeed, since $\mathcal{T}_{r}^{V}$ is conservative, i.e.,
$\mathcal{T}_{r}^{V}\mathbb{1}=\mathbb{1}$, we have

$\displaystyle 0<m_{U}\leq\overline{U}_{s}(w,h)\leq M_{U},\,(w,h)\in
H\times\mathbb{T},$

where $m_{U},M_{U}$ are the infimum and supremum of $U$. On the other hand,
one has

$\displaystyle\mathcal{T}_{t}^{V}\overline{U}_{s}-\overline{U}_{s}$
$\displaystyle=\frac{1}{s}\int_{0}^{s}\mathcal{T}_{t+r}^{V}Udr-\frac{1}{s}\int_{0}^{s}\mathcal{T}_{r}^{V}Udr$
$\displaystyle=\frac{1}{s}\int_{s}^{s+t}\mathcal{T}_{r}^{V}Udr-\frac{1}{s}\int_{0}^{t}\mathcal{T}_{r}^{V}Udr=\int_{0}^{t}\mathcal{T}_{r}^{V}\widehat{U}^{s}dr,$

where $\widehat{U}_{s}=\frac{\mathcal{T}_{s}^{V}U-U}{s}$. Since $U\in
C_{b}(H\times\mathbb{T})$, it follows from Lemma 4.4 that $\widehat{U}_{s}\in
C_{b}(H\times\mathbb{T})$. Hence $\overline{U}_{s}\in\mathcal{D}_{+}$ and the
claim is proved.

It then follows from (4.47) that

(4.48)
$\displaystyle\left\langle\overline{U}_{s},\mu_{1}\right\rangle=\left\langle\overline{U}_{s},\mu_{2}\right\rangle,\,\text{
for all }s>0.$

Again the fact $U\in C_{b}(H\times\mathbb{T})$ and Lemma 4.4 yield

$\lim_{s\to 0}\overline{U}_{s}(w,h)=U(w,h),\,(w,h)\in H\times\mathbb{T}.$

Therefore by Lebesgue’s dominated convergence theorem and (4.48), one has

$\displaystyle\left\langle U,\mu_{1}\right\rangle=\left\langle
U,\mu_{2}\right\rangle.$

Since $U\in C_{b}^{+}(H\times\mathbb{T})$ is arbitrary, we conclude
$\mu_{1}=\mu_{2}$.

Step 2: Proof of Item (2). We first claim that for any $U,\phi\in
C_{b}(H\times\mathbb{T})$ the trajectory $\phi_{t}:=\mathcal{R}_{t}^{V,U}\phi$
of the semigroup $\mathcal{R}_{t}^{V,U}$ defined as in (4.24) is the solution
to the following equation

(4.49)
$\displaystyle\phi_{t}=\mathcal{T}_{t}^{V}\phi+\int_{0}^{t}\mathcal{T}_{r}^{V}(U\phi_{t-r})dr.$

Note that by (4.20), (4.22) and (4.24), denoting
$\Xi_{s,t}^{V}=\exp\left(\int_{s}^{t}V(\mathcal{H}_{\tau})d\tau\right)$, and
$\mathbf{E}=\mathbf{E}_{(w,h)}$,

$\displaystyle\phi_{t}-\mathcal{T}_{t}^{V}\phi$
$\displaystyle=\mathcal{R}_{t}^{V,U}\phi-\mathcal{T}_{t}^{V}\phi$
$\displaystyle=\frac{1}{\lambda_{t,h,V}F^{V}}\mathbf{E}\left[\Xi_{0,t}^{V}(\Xi_{0,t}^{U}-1)(\phi
F^{V})(\mathcal{H}_{t})\right]$
$\displaystyle=\frac{1}{\lambda_{t,h,V}F^{V}}\mathbf{E}\left[\Xi_{0,t}^{V}\left(\int_{0}^{t}-\frac{d}{dr}\Xi_{r,t}^{U}dr\right)(\phi
F^{V})(\mathcal{H}_{t})\right]$
$\displaystyle=\frac{1}{\lambda_{t,h,V}F^{V}}\int_{0}^{t}\mathbf{E}\left[\Xi_{0,t}^{V}U(H_{r})\Xi_{r,t}^{U}(\phi
F^{V})(\mathcal{H}_{t})\right]dr.$

Therefore by the Markov property, and time homogeneity of
$\mathcal{P}_{t}^{V+U}$, as well as cocycle property of the eigenvalue
$\lambda_{t,h,V}$ from the integral representation (2.10), we deduce

$\displaystyle\phi_{t}-\mathcal{T}_{t}^{V}\phi$
$\displaystyle=\frac{1}{\lambda_{t,h,V}F^{V}}\int_{0}^{t}\mathbf{E}\left[\Xi_{0,r}^{V}U(H_{r})\mathbf{E}\left[\Xi_{r,t}^{V+U}(\phi
F^{V})(\mathcal{H}_{t})|\mathcal{F}_{r}\right]\right]dr$
$\displaystyle=\frac{1}{\lambda_{t,h,V}F^{V}}\int_{0}^{t}\mathbf{E}\left[\Xi_{0,r}^{V}U(H_{r})\mathbf{E}\left[\Xi_{r,t}^{V+U}(\phi
F^{V})(\mathcal{H}_{t})|\mathcal{F}_{r}\right]\right]dr$
$\displaystyle=\int_{0}^{t}\frac{1}{\lambda_{r,h,V}F^{V}}\mathbf{E}\left[\Xi_{0,r}^{V}U(H_{r})F^{V}(H_{r})\frac{1}{\lambda_{t-r,\beta_{r}h,V}F^{V}(H_{r})}\mathcal{P}_{t-r}^{V+U}(\phi
F^{V})(\mathcal{H}_{r})\right]dr$
$\displaystyle=\int_{0}^{t}\frac{1}{\lambda_{r,h,V}F^{V}}\mathbf{E}\left[\Xi_{0,r}^{V}U(H_{r})F^{V}(H_{r})\mathcal{R}_{t-r}^{V,U}(\mathcal{H}_{r})\right]dr$
$\displaystyle=\int_{0}^{t}\mathcal{T}_{r}^{V}(U\phi_{t-r})dr$

which completes the proof of the claim.

Now for any $\phi\in\mathcal{D}_{+}$, we have

$\displaystyle\mathcal{T}_{t}^{V}\phi(w,h)=\phi(w,h)+\int_{0}^{t}\mathcal{T}_{r}^{V}\mathbb{L}_{V}\phi(w,h)dr.$

Combining this and with $U_{\phi}=-\frac{\mathbb{L}_{V}\phi}{\phi}$ it is
straightforward to verify that $\phi_{t}\equiv\phi$ solves the equation

$\phi_{t}=\mathcal{T}_{t}^{V}\phi+\int_{0}^{t}\mathcal{T}_{r}^{V}(U_{\phi}\phi_{t-r})dr.$

Hence we have $\mathcal{R}_{t}^{V,U_{\phi}}\phi=\phi$ for all $t\geq 0$. Since
$\phi\in\mathcal{D}_{+}$, in view of (4.24), there are constants
$m_{\phi},M_{\phi}>0$ such that

$\displaystyle
m_{\phi}R_{0,t,h}^{V,U_{\phi}}\mathbb{1}_{H}(w)\leq\mathcal{R}_{t}^{V,U_{\phi}}\phi(w,h)\leq
M_{\phi}R_{0,t,h}^{V,U_{\phi}}\mathbb{1}_{H}(w).$

Also note

$\displaystyle\lim_{t\to\infty}\frac{1}{t}\log\mathcal{R}_{t}^{V,U_{\phi}}\phi(w,h)=\lim_{t\to\infty}\frac{1}{t}\log\phi(w,h)=0.$

Hence we obtain the desired Item (2)

$\displaystyle Q^{V}(U_{\phi})=\lim_{t\to\infty}\frac{1}{t}\log
R_{0,t,h}^{V,U_{\phi}}\mathbb{1}_{H}(w)=0.$

The proof is complete.

∎

## Appendix

##  A. A Kifer type criterion

The following Kifer type theorem was established in [23], which is based on
the work of Kifer [24] in the compact setting. Suppose that $X$ is a polish
space, $\Theta$ is a directed set and $\\{\zeta_{\theta},\theta\in\Theta\\}$
is a family of random probability measures on $X$ that are defined on some
probability spaces
$(\Omega_{\theta},\mathcal{F}_{\theta},\mathbf{P}_{\theta})$.
$r:\Theta\to\mathbb{R}$ is a given positive function such that
$\lim_{\theta\in\Theta}r(\theta)=+\infty$.

######  Theorem A.1 (Kifer Type Criterion [23]).

Assume that the family $\\{\zeta_{\theta},\theta\in\Theta\\}$ satisfies the
following two properties:

  1. (1)

The following limit exists for any $V\in C_{b}(X)$:

$Q(V)=\lim_{\theta\in\Theta}\frac{1}{r(\theta)}\log\int_{\Omega_{\theta}}\exp(r(\theta)\left\langle
V,\zeta_{\theta}\right\rangle)\mathbf{P}_{\theta}(d\omega).$

  2. (2)

There is a function $R:X\to[0,\infty]$ with compact level sets $\\{x\in
X:R(x)\leq a\\}$ for all $a\geq 0$ such that

$\int_{\Omega_{\theta}}\exp(r(\theta)\left\langle
R,\zeta_{\theta}\right\rangle)\mathbf{P}_{\theta}(d\omega)\leq
Ce^{cr(\theta)},\,\theta\in\Theta$

where $C$ and $c$ are positive constants.

Then $Q$ is $1$-Lipschitz and convex, and its Legendre transform $I$ given by

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

is a good rate function, for which the following upper bound holds for any
closed subset $F\in\mathcal{P}(X)$:

$\displaystyle\limsup_{\theta\in\Theta}\frac{1}{r(\theta)}\log\mathbf{P}_{\theta}\\{\zeta_{\theta}\in
F\\}\leq-\inf_{\mu\in F}I(\mu).$

If in addition, the following property is satisfied:

  1. (3)

Suppose there exists a vector space $\mathcal{V}\subset C_{b}(X)$ such that
the restrictions of its functions to any compact set $K\subset X$ form a dense
subspace in $C(K)$ and for any $V\in\mathcal{V}$ there is a unique equilibrium
state, i.e., a unique $\mu_{V}\in\mathcal{P}(H)$ such that

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

Then the following lower bound holds for any open subset $G\in\mathcal{P}(X)$:

$\displaystyle\liminf_{\theta\in\Theta}\frac{1}{r(\theta)}\log\mathbf{P}_{\theta}\\{\zeta_{\theta}\in
G\\}\geq-\inf_{\mu\in G}I(\mu).$

##  B. A priori estimates of solutions

Several standard estimates about the solution $\Phi_{0,t,h}(w_{0})$ of the
stochastic Navier-Stokes equation (2.2) with time symbol $h\in\mathbb{T}$ are
collected in the following Proposition B.1. Note that the constant $C$ in the
bounds does not depend on $h$ due to the boundedness of the periodic force
$f$. Let $\|f\|_{\infty}=\sup_{t\in\mathbb{R}}\|f\|$ and
$\mathcal{B}_{0}=\sum_{k=1}^{d}\|g_{k}\|^{2}$ for $f\in C_{b}(\mathbb{R},H)$
and $g_{k}\in H$.

######  Proposition B.1.

There is an $\eta_{0}=\eta_{0}(\nu,\|f\|_{\infty},\mathcal{B}_{0})$ so that
for any $\eta\in(0,\eta_{0}]$ and $h\in\mathbb{T}$, there is a constant $C>0$
independent of $h$ such that

  1. (1)

The following estimates hold:

(B.1)
$\displaystyle\mathbf{E}_{w_{0}}\exp\left(\eta\left\|\Phi_{0,t,h}\right\|^{2}\right)\leq
C\exp\left(\eta e^{-\nu t}\left\|w_{0}\right\|^{2}\right),$ (B.2)
$\displaystyle\mathbf{E}_{w_{0}}\exp\left(\eta\sup_{t\geq
0}\left(\left\|\Phi_{0,t,h}\right\|^{2}+\nu\int_{0}^{t}\left\|\Phi_{0,r,h}\right\|_{1}^{2}dr-
Ct\right)\right)\leq C\exp\left(\eta\left\|w_{0}\right\|^{2}\right),$ (B.3)
$\displaystyle\mathbf{E}_{w_{0}}\left[\|\Phi_{0,t,h}\|^{2}\exp\left(\eta\left\|\Phi_{0,t,h}\right\|^{2}\right)\right]\leq
C(t)\exp\left(\eta\left\|w_{0}\right\|^{2}\right)$

  2. (2)

For any $h_{1},h_{2}\in\mathbb{T}$, we have

(B.4)
$\displaystyle\mathbf{E}_{w_{0}}\left\|\Phi_{0,t,h_{1}}-\Phi_{0,t,h_{2}}\right\|^{2}\leq
Ce^{rt}\exp\left({\eta\|w_{0}\|^{2}}\right)\sup_{t\in\mathbb{R}}\|f(\beta_{t}h_{1})-f(\beta_{t}h_{2})\|^{2}.$

  3. (3)

For any $R>0$ and $w_{1},w_{2}\in B_{R}^{H}$, we have

(B.5)
$\displaystyle\mathbf{E}\|\Phi_{0,t,h}(w_{1})-\Phi_{0,t,h}(w_{2})\|^{2}\leq
C\|w_{1}-w_{2}\|^{2}e^{\eta R^{2}+rt}.$

  4. (4)

For any $m\geq 1$,

(B.6) $\displaystyle\begin{aligned}
&\mathbf{E}_{w_{0}}\|\Phi_{0,t,h}\|^{2m}\leq e^{-\nu
mt}\|w_{0}\|^{2m}+C(m),\\\ &\mathbf{E}_{w_{0}}\|\Phi_{0,t,h}\|_{2}^{2m}\leq
C(m,t)(1+\|w_{0}\|^{12m}).\end{aligned}$

Estimates (B.1) and (B.2) can be proved as in [19]. Inequalities (B.4) and
(B.5) can be proved as in [20]. One can prove (B.3) and (B.6) as [30]. See
also [26].

##  C. Growth condition, uniform irreducibility and uniform Feller property

###  C.1. The Growth Condition

The following growth condition is useful when working with the non-
conservative Feynman-Kac operator $P^{V}_{0,t,h}$ in a non-compact phase
space, which was first introduced in [23] in the time homogeneous setting.

######  Proposition C.1.

Assume $V\in B_{b}(H\times\mathbb{T})$. Recall that
$\mathfrak{m}_{\eta}(w)=e^{\eta\|w\|^{2}}$ and
$\mathfrak{m}_{q}(w)=1+\|w\|^{2q}$. Then there are $R_{0}>0$ and $q=q(V)\geq
1$ such that

$\displaystyle M_{\eta}:=\sup_{t\geq
0,h\in\mathbb{T}}\frac{\|P_{0,t,h}^{V}\mathfrak{m}_{\eta}\|_{L_{\mathfrak{m}_{\eta}}^{\infty}(H)}}{\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}<\infty.$
$\displaystyle M_{q}:=\sup_{t\geq
0,h\in\mathbb{T}}\frac{\|P_{0,t,h}^{V}\mathfrak{m}_{q}\|_{L_{\mathfrak{m}_{q}}^{\infty}(H)}}{\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}<\infty.$
$\displaystyle M_{H_{2}}:=\sup_{t\geq
0,h\in\mathbb{T}}\frac{\|P_{0,t,h}^{V}N_{2}\|_{L_{\mathfrak{m}_{\eta}}^{\infty}(H)}}{\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}<\infty,\quad
N_{2}(w)=\|w\|_{2}^{2},$

for all $\eta\in(0,\eta_{0}]$. Here $\|\cdot\|_{R_{0}}$ is the $L^{\infty}$
norm on $B_{R_{0}}^{H_{2}}$. In particular, we have

$\displaystyle P_{0,t,h}^{V}\mathfrak{m}_{\eta}(w)\leq
M_{\eta}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\mathfrak{m}_{\eta}(w),\,\text{
for all }t\geq 0,h\in\mathbb{T},w\in H.$

The proof of Proposition C.1 is based on [30] where the time homogeneous case
was proved, which is based on the following hyper-exponential recurrence
property of the solutions to the Navier-Stokes system, see also [23, 26].

######  Proposition C.2.

For any $R>0,h\in\mathbb{T}$, let $\tau_{h}(R)$ be the first hitting time of
the solution $\Phi_{0,t,h}$ to the set $B_{R}^{H_{2}}$:

$\displaystyle\tau_{h}(R)=\inf\left\\{t\geq 0:\Phi_{0,t,h}\in
B_{R}^{H_{2}}\right\\}.$

Then for any $\gamma>0$, there is $q_{0}=q_{0}(\gamma)>\gamma/\nu$ so that for
any $q\geq q_{0}$, there are positive numbers $R,C$ depending on $q$ such that

(C.1)
$\displaystyle\sup_{h\in\mathbb{T}}\mathbf{E}_{w}\exp\left(\gamma\tau_{h}(R)\right)\leq
C\mathfrak{m}_{q}(w),\,w\in H.$

Here $\nu>0$ is the viscosity.

For the sake of completeness, we provide proofs for both propositions in the
current time inhomogeneous setting. The translation identity (2.6) and uniform
in $h$ bounds from Proposition B.1 are the crucial ingredients when dealing
with time inhomogeneity throughout the proof. We first give the proof of
Proposition C.1 based on Proposition C.2, whose proof is given at the end of
this subsection.

###### Proof of Proposition C.1.

Since we are estimating the ration of $P_{0,t,h}^{V}$, we can assume

$m_{V}=\inf_{(w,h)\in H\times\mathbb{T}}V(w,h)\geq 0.$

Otherwise we can replace it by $V-m_{V}$. The proof is divided into four
steps.

Step 1: Proof of $M_{q}<\infty$ when $\mathfrak{m}_{q}$ is replaced by
$\mathbb{1}_{H}$. Here we aim to show the existence of $q_{0},R_{0}\geq 1$
such that

(C.2) $\displaystyle\sup_{t\geq
0,h\in\mathbb{T}}\frac{\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{L_{\mathfrak{m}_{q}}^{\infty}(H)}}{\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}}<\infty.$

In view of Proposition C.2, we pick any $q,R_{0}$ the numbers corresponding to
$\gamma=\|V\|_{\infty}$, with $\tau_{h}(R_{0})$ be the first hitting time of
the ball $B_{R_{0}}^{H_{2}}$. Now we write

(C.3) $\displaystyle
P_{0,t,h}^{V}\mathbb{1}_{H}(w)=\mathbf{E}_{(w,h)}\Xi_{t}^{V}=\mathbf{E}_{(w,h)}\mathbb{1}_{G_{t,h}}\Xi_{t}^{V}+\mathbf{E}_{(w,h)}\mathbb{1}_{G_{t,h}^{c}}\Xi_{t}^{V}:=I_{1}+I_{2},$

where $\Xi_{t}^{V}=\exp\left(\int_{0}^{t}V(H_{r})dr\right)$ and
$G_{t,h}=\left\\{\tau_{h}(R_{0})>t\right\\}$. Since $V\geq 0$ and
$\tau_{h}(R_{0})>t$ on $G_{t,h}$, we obtain

(C.4) $\displaystyle
I_{1}\leq\mathbf{E}_{(w,h)}\Xi_{\tau_{h}(R_{0})}^{V}\leq\mathbf{E}_{w}\exp\left(\gamma\tau_{h}(R_{0})\right)\leq
C\mathfrak{m}_{q}(w)\leq
C\mathfrak{m}_{q}(w)\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}},$

by Proposition C.2. By the strong Markov inequality and the fact that
$\Phi_{0,\tau_{h}(R_{0}),h}\in B_{R_{0}}^{H_{2}}$, we have

(C.5)
$\displaystyle\begin{split}I_{2}&=\mathbf{E}_{(w,h)}\left[\mathbb{1}_{G_{t,h}^{c}}\Xi_{\tau_{h}(R_{0})}^{V}\mathbf{E}_{(w,h)}\left[\exp\left(\int_{\tau_{h}(R_{0})}^{t}V(H_{r})dr\right)\Big{|}\mathcal{F}_{\tau_{h}(R_{0})}\right]\right]\\\
&=\mathbf{E}_{(w,h)}\left[\mathbb{1}_{G_{t,h}^{c}}\Xi_{\tau_{h}(R_{0})}^{V}P_{\tau_{h}(R_{0}),t,h}^{V}\mathbb{1}_{H}(\Phi_{0,\tau_{h}(R_{0}),h})\right]\\\
&\leq\mathbf{E}_{w}\left[\mathbb{1}_{G_{t,h}^{c}}\exp(\gamma\tau_{h}(R_{0}))P_{0,t,h}^{V}\mathbb{1}_{H}(\Phi_{0,\tau_{h}(R_{0}),h})\right]\\\
&\leq\mathbf{E}_{w}\left[\exp(\gamma\tau_{h}(R_{0}))\right]\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\leq
C\mathfrak{m}_{q}(w)\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}},\end{split}$

by Proposition C.2 again. By combining (C.3)-(C.5) we obtain the desired
(C.2).

The proof of Step 1 is complete.

Step 2: Proof of $M_{q}<\infty$. We first show the estimate for discrete time
and then pass to the continuous time case by evolution property. Noting that
$\delta:=e^{\gamma-\nu q}<1$ by Proposition C.2, hence by translation identity
(2.6) one has

$\displaystyle P_{0,k,h}^{V}\mathfrak{m}_{q}(w)$
$\displaystyle=P_{0,k-1,h}^{V}\left(P_{0,1,\beta_{k-1}h}^{V}\mathfrak{m}_{q}\right)(w)$
$\displaystyle\leq
P_{0,k-1,h}^{V}\left(e^{\gamma}P_{0,1,\beta_{k-1}h}\mathfrak{m}_{q}\right)(w)\leq
e^{\gamma}\left(e^{-\nu
q}P_{0,k-1,h}^{V}\mathfrak{m}_{q}(w)+C(q)P_{0,k-1,h}^{V}\mathbb{1}_{H}\right)$
$\displaystyle\leq\delta
P_{0,k-1,h}^{V}\mathfrak{m}_{q}(w)+C(q,\gamma)P_{0,k,h}^{V}\mathbb{1}_{H}(w)$

where we used estimate (B.6) in the penultimate step and monotonicity of
$P_{0,k,h}^{V}\mathbb{1}_{H}$ in $k$ in the last step. Iterating this
inequality we obtain

$\displaystyle
P_{0,k,h}^{V}\mathfrak{m}_{q}(w)\leq\delta^{k}\mathfrak{m}_{q}(w)+\frac{C}{1-\delta}P_{0,k,h}^{V}\mathbb{1}_{H}(w),$

where the constant $C$ does not depend on $w,h,k$. Combining this with (C.2),
we obtain $M_{q}<\infty$ in the discrete time case.

For the continuous time case, denoting $[t]$ the integer part of $t$, by (B.6)
and (2.6), we have

$\displaystyle
P^{V}_{[t],t,h}\mathfrak{m}_{q}(w)=P^{V}_{0,t-[t],\beta_{[t]}h}\mathfrak{m}_{q}(w)\leq
e^{\gamma}P_{0,t-[t],\beta_{[t]}h}\mathfrak{m}_{q}(w)\leq
e^{\gamma}\mathfrak{m}_{q}(w)+C,\,w\in H,h\in\mathbb{T}.$

Hence

$\displaystyle P^{V}_{0,t,h}\mathfrak{m}_{q}(w)$
$\displaystyle=P^{V}_{0,[t],h}\left(P^{V}_{[t],t,h}\mathfrak{m}_{q}\right)(w)\leq
e^{\gamma}P^{V}_{0,[t],h}\mathfrak{m}_{q}(w)+CP^{V}_{0,[t],h}\mathbb{1}_{H}(w)$
$\displaystyle\leq
C\mathfrak{m}_{q}(w)\|P_{0,[t],h}^{V}\mathbb{1}_{H}\|_{R_{0}}\leq
C\mathfrak{m}_{q}(w)\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}$

by the proved discrete time case, estimate (C.2), and monotonicity of
$t\to\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}$. Therefore $M_{q}<\infty$ in the
continuous time as well.

The proof of Step 2 is complete.

Step 3: Proof of $M_{\eta}<\infty$. As in the proof of the previous step, we
only need to show the discrete time case. The passage to the continuous time
is similar by invoking (B.1).

Let $A>0$ and note that

(C.6)
$\displaystyle\begin{split}P_{0,k,h}^{V}\mathfrak{m}_{\eta}(w)&=\mathbf{E}_{(w,h)}\left[\mathbb{1}_{\|\Phi_{0,k,h}\|^{2}\leq
A}\Xi_{k}^{V}\mathfrak{m}_{\eta}(\Phi_{0,k,h})\right]+\mathbf{E}_{(w,h)}\left[\mathbb{1}_{\|\Phi_{0,k,h}\|^{2}\geq
A}\Xi_{k}^{V}\mathfrak{m}_{\eta}(\Phi_{0,k,h})\right]\\\
&:=I_{k,h}+J_{k,h}.\end{split}$

It follow from (C.2) that

$\displaystyle P_{0,k,h}^{V}\mathbb{1}_{H}(w)\leq
C\mathfrak{m}_{q}(w)\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\leq
C\mathfrak{m}_{\eta}(w)\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}},\,(w,h)\in
H\times\mathbb{T}.$

Therefore

(C.7) $\displaystyle I_{k,h}\leq e^{\eta A}P_{0,k,h}^{V}\mathbb{1}_{H}(w)\leq
C\mathfrak{m}_{\eta}(w)\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}},$

where $C$ is independent of $k,h,w$. By (2.6), (B.3) and denoting
$F(w)=\|w\|^{2}\mathfrak{m}_{\eta}(w)$ for $w\in H$, we have

(C.8) $\displaystyle\begin{split}J_{k,h}\leq
A^{-1}P_{0,k,h}^{V}F(w)&=A^{-1}P_{0,k-1,h}^{V}\left(P_{0,1,\beta_{k-1}h}^{V}F\right)(w)\\\
&\leq e^{\gamma}A^{-1}P_{0,k-1,h}^{V}\left(P_{0,1,\beta_{k-1}h}F\right)(w)\\\
&\leq A^{-1}CP_{0,k-1,h}^{V}\mathfrak{m}_{\eta}(w).\end{split}$

Combining (C.6)-(C.8) and choosing $A$ large such that $\delta:=A^{-1}C<1$, we
obtain

$\displaystyle P_{0,k,h}^{V}\mathfrak{m}_{\eta}(w)\leq
C\mathfrak{m}_{\eta}(w)\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}}+\delta
P_{0,k-1,h}^{V}\mathfrak{m}_{\eta}(w).$

By iteration, one has

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

Hence $M_{\eta}<\infty$.

The proof of Step 3 is complete.

Step 4: Proof of $M_{H_{2}}<\infty$. This follows from (B.6) and
$M_{\eta}<\infty$ proved in the previous step. Indeed, since
$\mathfrak{m}_{q}\leq C\mathfrak{m}_{\eta}$, one has

$\displaystyle P_{0,k,h}^{V}N_{2}(w)$
$\displaystyle=P_{0,k-1,h}^{V}(P_{0,1,\beta_{k-1}h}^{V}N_{2})(w)\leq
e^{\gamma}P_{0,k-1,h}^{V}(P_{0,1,\beta_{k-1}h}N_{2})(w)$ $\displaystyle\leq
CP_{0,k-1,h}^{V}\mathfrak{m}_{\eta}(w)\leq
C\mathfrak{m}_{\eta}(w)\|P_{0,k-1,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\leq
C\mathfrak{m}_{\eta}(w)\|P_{0,k,h}^{V}\mathbb{1}_{H}\|_{R_{0}}.$

Hence $M_{H_{2}}<\infty$, which finishes the proof of Step 4.

The proof of Proposition C.1 is complete. ∎

###### Proof of Proposition C.2.

The proof is divided into two steps. First we prove hyper-exponential
recurrence in $H$ and then we show the desired recurrence in $H_{2}$.

Step 1: Hyper-exponential recurrence in $H$. For any $r>0$, denote
$\overline{\tau}_{h}(r)$ the first hitting time

$\displaystyle\overline{\tau}_{h}(r)=\inf\left\\{t\geq 0:\Phi_{0,t,h}\in
B_{r}^{H}\right\\}.$

We aim to show that for any $\gamma>0$, there is $q_{0}=q_{0}(\gamma)$ so that
for any $q\geq q_{0}$, there are positive numbers $r,\overline{C}$ depending
on $q$ such that

(C.9)
$\displaystyle\sup_{h\in\mathbb{T}}\mathbf{E}_{w}\exp\left(\gamma\overline{\tau}_{h}(r)\right)\leq\overline{C}\mathfrak{m}_{q}(w),\,w\in
H.$

Denote by $N_{0}^{2q}(w)=\|w\|^{2q},\,w\in H$. It follows from (B.6) that

(C.10) $\displaystyle
P_{0,1,h}N_{0}^{2q}(w)=\mathbf{E}_{w}\|\Phi_{0,1,h}\|^{2q}\leq\delta(\|w\|^{2q}\vee
r),\,(w,h)\in H\times\mathbb{T},$

where $\delta=e^{-\nu q}<1$ and $r=e^{\nu q}C>1$, with the constant $C=C(q)$
from (B.6). Let

$\displaystyle
R_{k,h}(w)=\mathbf{E}_{w}\left(\mathbb{1}_{\\{\overline{\tau}_{h}(r)>k\\}}\|\Phi_{0,k,h}\|^{2q}\right).$

Note that
$\mathbb{1}_{\\{\overline{\tau}_{h}(r)>k\\}}\geq\mathbb{1}_{\\{\overline{\tau}_{h}(r)>k+1\\}}$
and on the event $\\{\overline{\tau}_{h}(r)>k\\}$ one has
$\|\Phi_{0,k,h}\|^{2q}>r^{2q}>r$. Therefore by Markov inequality, (2.6) and
(C.11), we have

$\displaystyle R_{k+1,h}(w)$
$\displaystyle\leq\mathbf{E}_{w}\left(\mathbb{1}_{\overline{\tau}_{h}(r)>k}\|\Phi_{0,k+1,h}\|^{2q}\right)=\mathbf{E}_{w}\left(\mathbb{1}_{\overline{\tau}_{h}(r)>k}\mathbf{E}_{w}\left[\|\Phi_{0,k+1,h}\|^{2q}|\mathcal{F}_{k}\right]\right)$
$\displaystyle=\mathbf{E}_{w}\left(\mathbb{1}_{\overline{\tau}_{h}(r)>k}P_{k,k+1,h}N_{0}^{2q}(\Phi_{0,k,h})\right)=\mathbf{E}_{w}\left(\mathbb{1}_{\overline{\tau}_{h}(r)>k}P_{0,1,\beta_{k}h}N_{0}^{2q}(\Phi_{0,k,h})\right)$
$\displaystyle\leq\delta\mathbf{E}_{w}\left(\mathbb{1}_{\overline{\tau}_{h}(r)>k}\|\Phi_{0,k,h}\|^{2q}\right)=\delta
R_{k,h}(w).$

By iteration, one has

$\displaystyle R_{k,h}(w)\leq\delta^{k}\|w\|^{2q},$

which in turn implies

$\displaystyle\mathbf{P}_{w}(\overline{\tau}_{h}(r)>k)\leq
r^{-2q}R_{k,h}(w)\leq\delta^{k}r^{-2q}\|w\|^{2q},\,k\geq 0,w\in H.$

Hence for any $\gamma>0$, by choosing $q>\gamma/\nu$ large such that
$e^{\gamma}\delta=e^{\gamma-\nu q}<1$, we obtain the desired (C.9):

(C.11)
$\displaystyle\begin{split}\mathbf{E}_{w}\exp\left(\gamma\overline{\tau}_{h}(r)\right)&\leq
1+\sum_{k=1}^{\infty}e^{\gamma k}\mathbf{P}_{w}(\overline{\tau}_{h}(r)>k-1)\\\
&\leq 1+r^{-2q}\|w\|^{2q}\sum_{k=1}^{\infty}e^{\gamma k}\delta^{k-1}\leq
C\mathfrak{m}_{q}(w),\end{split}$

for some constant $C$ independent of $h\in\mathbb{T}$.

Step 2: Hyper-exponential recurrence in $H_{2}$. First it follows from (B.6)
and Markov inequality that for any $p\in(0,1)$, there is a large $R>0$ such
that

(C.12) $\displaystyle P_{0,1,h}\mathbb{1}_{H\setminus
B_{R}^{H_{2}}}(w)=\mathbf{P}_{w}\left(\|\Phi_{0,1,h}\|_{2}\geq R\right)\leq
R^{-2}\mathbf{E}_{w}\|\Phi_{0,1,h}\|_{2}^{2}\leq p,$

for all $h\in\mathbb{T}$ and $w\in B_{r}^{H}$, where $r$ is the radius from
(C.9) with corresponding $\overline{\tau}_{h}(r)$. Define recursively the
following sequence of stopping times

$\displaystyle\tau_{0,h}^{\prime}=\overline{\tau}_{h}(r),\,\tau_{n,h}^{\prime}=\inf\left\\{t\geq\tau_{n-1,h}^{\prime}+1:\Phi_{0,t,h}\in
B_{r}^{H}\right\\},\,\text{ and }\tau_{n,h}=\tau_{n,h}^{\prime}+1,\,\text{ for
}n\geq 1.$

Note that $\tau_{0,h}^{\prime}$ is the first hitting time of the ball
$B_{r}^{H}$, and $\tau_{n,h}^{\prime}$ is the first recurrence time after
$\tau_{n-1,h}^{\prime}+1$. Also let

$\displaystyle\hat{n}=\min\left\\{n\geq 0:\Phi_{0,\tau_{n,h},h}\in
B_{R}^{H_{2}}\right\\}$

be the first time the solution entering the ball $B_{R}^{H_{2}}$ among the
sequence $\tau_{n,h}$. Note that for any $k,M\geq 1$, since
$\left\\{\tau_{h}(R)>\tau_{k,h}\right\\}\subset\left\\{\hat{n}>k\right\\}$, we
have

(C.13) $\displaystyle\begin{split}\mathbf{P}_{w}(\tau_{h}(R)\geq
M)&=\mathbf{P}_{w}(\tau_{h}(R)\geq
M,\tau_{k,h}<M)+\mathbf{P}_{w}(\tau_{h}(R)\geq M,\tau_{k,h}\geq M)\\\
&\leq\mathbf{P}_{w}(\tau_{h}(R)>\tau_{k,h})+\mathbf{P}_{w}(\tau_{k,h}\geq
M)\\\ &\leq\mathbf{P}_{w}(\hat{n}>k)+\mathbf{P}_{w}(\tau_{k,h}\geq
M).\end{split}$

In what follows we estimate the probabilities on the right hand side.

It follows from translation identity (2.6), estimate (C.12), the fact
$\Phi_{0,\tau_{k,h}^{\prime},h}\in B_{r}^{H}$ and the strong Markov property
that

$\displaystyle\mathbf{P}_{w}(\hat{n}>k)$
$\displaystyle=\mathbf{E}_{w}\left[\mathbb{1}_{\left\\{\hat{n}>k-1\right\\}}\mathbb{1}_{H\setminus
B_{R}^{H_{2}}}(\Phi_{0,\tau_{k,h},h})\right]$
$\displaystyle=\mathbf{E}_{w}\left[\mathbb{1}_{\left\\{\hat{n}>k-1\right\\}}\mathbf{E}_{w}\left[\mathbb{1}_{H\setminus
B_{R}^{H_{2}}}(\Phi_{0,\tau_{k,h},h})|\mathcal{F}_{\tau_{k,h}^{\prime}}\right]\right]$
$\displaystyle=\mathbf{E}_{w}\left[\mathbb{1}_{\left\\{\hat{n}>k-1\right\\}}P_{\tau_{k,h}^{\prime},\tau_{k,h},h}\mathbb{1}_{H\setminus
B_{R}^{H_{2}}}(\Phi_{0,\tau_{k,h}^{\prime},h})\right]$
$\displaystyle=\mathbf{E}_{w}\left[\mathbb{1}_{\left\\{\hat{n}>k-1\right\\}}P_{0,1,\beta_{\tau_{k,h}^{\prime}}h}\mathbb{1}_{H\setminus
B_{R}^{H_{2}}}(\Phi_{0,\tau_{k,h}^{\prime},h})\right]\leq
p\mathbf{E}_{w}\left[\mathbb{1}_{\left\\{\hat{n}>k-1\right\\}}\right]=p\mathbf{P}_{w}(\hat{n}>k-1),$

which, by iteratively using the strong Markov property, implies

(C.14) $\displaystyle\mathbf{P}_{w}(\hat{n}>k)\leq p^{k},\,\text{ for }w\in
H,k\geq 0.$

In a similar fashion, by Markov’s inequality, the recurrence (C.9) with
constant $\overline{C}$, and the strong Markov property, we have for any
$M\geq 1$,

$\displaystyle\mathbf{P}_{w}(\tau_{k,h}\geq M)$ $\displaystyle\leq e^{-\gamma
M}\mathbf{E}_{w}\exp(\gamma\tau_{k,h})=e^{-\gamma
M}\mathbf{E}_{w}\left[\mathbf{E}_{w}\left[\exp(\gamma(\tau_{k,h}^{\prime}+1))|\mathcal{F}_{\tau_{k-1,h}}\right]\right]$
$\displaystyle=e^{-\gamma
M}e^{\gamma}\mathbf{E}_{w}\left[\exp(\gamma\tau_{k-1,h})\mathbf{E}_{\Phi_{0,\tau_{k-1,h},h}}\exp(\gamma\overline{\tau}_{\beta_{\tau_{k-1,h}}h}(r))\right]$
$\displaystyle\leq e^{-\gamma
M}\overline{C}e^{\gamma}\mathbf{E}_{w}\left[\exp(\gamma\tau_{k-1,h})\mathfrak{m}_{q}(\Phi_{0,\tau_{k-1,h},h})\right]$
$\displaystyle=e^{-\gamma
M}\overline{C}e^{\gamma}\mathbf{E}_{w}\left[\exp(\gamma\tau_{k-1,h})\mathbf{E}_{\Phi_{0,\tau_{k-1,h}^{\prime},h}}\mathfrak{m}_{q}(\Phi_{0,\tau_{k-1,h},h})\right]$
$\displaystyle\leq e^{-\gamma M}\overline{C}e^{\gamma}(e^{-\nu
q}r^{2q}+C(q))\mathbf{E}_{w}\exp(\gamma\tau_{k-1,h}),$

where we used the fact that $\tau_{k-1,h}$ is
$\mathcal{F}_{\tau_{k-1,h}^{\prime}}$ measurable in the last equality, and
(B.6) with $\Phi_{0,\tau_{k-1,h}^{\prime},h}\in B_{r}^{H}$ in the last
inequality. Hence by iteratively using the strong Markov inequality, we obtain

(C.15) $\displaystyle\mathbf{P}_{w}(\tau_{k,h}\geq M)\leq e^{-\gamma
M}C^{k}\mathfrak{m}_{q}(w),\,w\in H,h\in\mathbb{T},$

with $C=\overline{C}e^{\gamma}(e^{-\nu q}r^{2q}+C(q))$ independent of $w,M,h$.

Now combining (C.13)-(C.15), one has

$\displaystyle\mathbf{P}_{w}(\tau_{h}(R)\geq M)\leq p^{k}+e^{-\gamma
M}C^{k}\mathfrak{m}_{q}(w).$

Choosing $k=\frac{\gamma M}{2\ln C}$ and $R$ large such that $p=1/C$ by
(C.12), then we have

(C.16) $\displaystyle\mathbf{P}_{w}(\tau_{h}(R)\geq M)\leq 2e^{-\gamma
M/2}\mathfrak{m}_{q}(w)$

for any $\gamma>0,M\geq 1$ and $(w,h)\in H\times\mathbb{T}$. This implies the
desired hyper-exponential recurrence (C.1) by following the same reasoning as
in (C.11). Note that to obtain (C.1) with any $\gamma>0$ we can replace the
$\gamma$ in (C.16) by $4\gamma$.

The proof of Proposition C.2 is complete. ∎

###  C.2. Uniform Irreducibility

In this subsection we prove the uniform irreducibility, which follows from the
well-known controllability results of Agrachev and Sarychev [1, 2, 15], the
cocycle property (2.7) and standard compactness arguments.

######  Proposition C.3 (Uniform Irreducibility ).

Let $V\in B_{b}(H\times\mathbb{T})$. Given any $\rho,r>0,R\geq 1$, as long as
$\ell\geq 2\ln R$, then there is $p=p(V,\rho,r,\ell)>0$ such that

(C.17) $\displaystyle\inf_{(w_{0},h,w)\in B_{R}^{H}\times\mathbb{T}\times
B_{\rho}^{H_{2}}}P_{0,\ell,h}^{V}(w_{0},B^{H,H_{2}}_{r}(w))\geq p$

Moreover, there is a large $L_{0}=L_{0}(\ell,R)$ such that for any $L\geq
L_{0}$, there holds

(C.18) $\displaystyle\inf_{(w_{0},h,w)\in B_{R}^{H}\times\mathbb{T}\times
B_{\rho}^{H_{2}}}P_{0,\ell,h}^{V}(w_{0},B^{H,B_{L}^{H_{2}}}_{r}(w))\geq p/2.$

If $\displaystyle m_{V}=\inf_{w\in H}V(w)\geq 0$, then one can choose
$p=p(\rho,r)$ that is independent of $V,\ell$.

###### Proof.

First of all, we claim the existence a large time $t_{0}$ depending on $R$ and
a large number $r_{0}>0$ independent of $w_{0},R,h,t_{0}$, such that

(C.19) $\displaystyle\inf_{(w_{0},h)\in
B^{H}_{R}\times\mathbb{T}}P_{0,t_{0},h}(w_{0},B_{r_{0}}^{H_{2}})\geq 1/2.$

Let $N(w)=\|w\|^{2}$ and $N_{2}(w)=\|w\|_{2}^{2}$. By estimate (B.6), we have
for any $t\geq 1$, and $w_{0}\in B^{H}_{R}$,

$\displaystyle P_{0,t,h}N_{2}(w_{0})$
$\displaystyle=P_{0,t-1,h}(P_{0,1,\beta_{t-1}h}N_{2})(w_{0})\leq
C\left(P_{0,t-1,h}N^{6}(w_{0})+1\right)$ $\displaystyle\leq
C(e^{-6(t-1)}\|w_{0}\|^{12}+1)\leq C(e^{-6(t+1)}R^{12}+1)$

where $C$ is a constant independent of $t,h,R$. Since $P_{0,t,h}(w_{0},\cdot)$
is concentrated on $H_{2}$, by choosing $t_{0}$ large so that
$e^{-6(t_{0}+1)}R^{12}\leq 1$ and choosing a constant $r_{0}>2\sqrt{C}$, we
have by Markov inequality that

(C.20) $\displaystyle P_{0,t_{0},h}(w_{0},B_{r_{0}}^{H_{2}})$
$\displaystyle=1-P_{0,t_{0},h}(w_{0},H_{2}\setminus B_{r_{0}}^{H_{2}})\geq
1-\frac{(P_{0,t_{0},h}N_{2}(w_{0}))^{2}}{r_{0}^{2}}\geq 1/2.$

Next we show the existence of $p_{0}>0$, depending only on $r_{0},r,\rho$ such
that

(C.21) $\displaystyle\inf_{(w_{0},h,w)\in
B_{r_{0}}^{H_{2}}\times\mathbb{T}\times
B_{\rho}^{H_{2}}}P_{0,1,h}(w_{0},B^{H,H_{2}}_{r}(w))\geq p_{0}.$

Let

$\Omega_{1}=\\{X\in C([0,1],\mathbb{R}^{d}):X(0)=0\\}$

be the restriction of $\Omega$ on $[0,1]$. In view of the controllability
results of Agrachev and Sarychev [1, 2, 15] (see [27] for a verification in
the time inhomogeneous case), the Navier-Stokes system (2.1) is approximately
controllable. That is, for any $(w_{0},v,h)\in H^{2}\times\mathbb{T}$ and
$r>0$, if we replace $f(t,x)$ by $f(\beta_{t}h,x)$ in $\eqref{NS}$, then there
is a piecewise smooth map $X=X_{w_{0},v,h,r}\in\Omega_{1}$ such that
$\Phi_{0,1,h}(w_{0},X)\in\mathring{B}_{r/4}^{H}(v),$ where
$\Phi_{0,t,h}(w_{0},X)$ is the solution of (2.1) with symbol $h$ and $W(t)$
replaced by $X(t)$. Since $\Phi_{0,1,h}(w_{0},\cdot):\Omega_{1}\to H$ is
continuous, there is an $\varepsilon=\varepsilon(w_{0},v,h,r)>0$ such that for
any $Y\in\mathring{B}_{\varepsilon}^{\Omega_{1}}(X)$ one has

$\|\Phi_{0,1,h}(w_{0},Y)-\Phi_{0,1,h}(w_{0},X)\|<r/4.$

Hence by triangle inequality and regularizing property of the Navier-Stokes
systems, we obtain that $Y\in\mathring{B}_{\varepsilon}^{\Omega_{1}}(X)$
implies $\Phi_{0,1,h}(w_{0},Y)\in\mathring{B}_{r/2}^{H,H_{2}}(v)$.

Since $B^{H_{2}}_{\rho}$ is compact in $H$, its open cover
$\left\\{\mathring{B}_{r/2}^{H,H_{2}}(v)\right\\}_{v\in B^{H_{2}}_{\rho}}$ has
a finite subcover
$\left\\{\mathring{B}_{r/2}^{H,H_{2}}(v_{k})\right\\}_{k=1}^{K}$ where the
positive integer $K=K(r,\rho)$. Let
$\varepsilon_{k}=\varepsilon(w_{0},v_{k},h,r)$ and $X_{k}$ be the path from
the controllability result corresponding to $v_{k}$. Note that
$\mathbf{P}(\mathring{B}_{\varepsilon_{k}}^{\Omega_{1}}(X_{k}))>0$ for each
$1\leq k\leq K$ since the Wiener measure has a full support. Hence

$p(w_{0},h,r,\rho):=\min_{1\leq k\leq
K}\mathbf{P}(\mathring{B}_{\varepsilon_{k}}^{\Omega_{1}}(X_{k}))>0.$

Now for any $w\in B^{H_{2}}_{\rho}$, there is some $v_{k}$ such that
$w\in\mathring{B}_{r/2}^{H,H_{2}}(v_{k})$. Therefore
$\mathring{B}_{r/2}^{H,H_{2}}(v_{k})\subset B^{H,H_{2}}_{r}(w)$. It then
follows that

$\displaystyle P_{0,1,h}(w_{0},B^{H,H_{2}}_{r}(w))\geq
P_{0,1,h}(w_{0},\mathring{B}_{r/2}^{H,H_{2}}(v_{k}))\geq\mathbf{P}(\mathring{B}_{\varepsilon_{k}}^{\Omega_{1}}(X_{k}))\geq
p(w_{0},h,r,\rho),\quad(w_{0},h)\in H\times\mathbb{T}.$

This combined with a continuity and compactness argument similar to the proof
of Lemma 4.4 in [27] implies the existence of a number
$p_{0}=p_{0}(r_{0},r,\rho)>0$ such that

$\inf_{(w_{0},h)\in
B_{r_{0}}^{H_{2}}\times\mathbb{T}}P_{0,1,h}(w_{0},B^{H,H_{2}}_{r}(w))\geq
p_{0},$

where $B_{r_{0}}^{H_{2}}$ is from (C.20).

Lastly, it follows from the Chapman-Kolmogorov relation and inequalities
(C.19) and (C.21) that

(C.22)
$\displaystyle\begin{split}P_{0,t_{0}+1,h}(w_{0},B^{H,H_{2}}_{r}(w))&=\int_{H}P_{0,1,\beta_{t_{0}}h}(u,B^{H,H_{2}}_{r}(w))P_{0,t_{0},h}(w_{0},du)\\\
&\geq\int_{B_{r_{0}}^{H_{2}}}P_{0,1,\beta_{t_{0}}h}(u,B^{H,H_{2}}_{r}(w))P_{0,t_{0},h}(w_{0},du)\geq
p_{0}/2\end{split}$

for any $(w_{0},h,w)\in B_{R}^{H}\times\mathbb{T}\times B_{\rho}^{H_{2}}$.
Hence

$P^{V}_{0,t_{0}+1,h}(w_{0},B^{H,H_{2}}_{r}(w))\geq
e^{(t_{0}+1)m_{V}}P_{0,t_{0}+1,h}(w_{0},B^{H,H_{2}}_{r}(w))\geq\frac{p_{0}e^{(t_{0}+1)m_{V}}}{2}.$

The inequality (C.17) then follows with $\ell=t_{0}+1\geq 2\ln R$ and
$p=\frac{p_{0}e^{(t_{0}+1)m_{V}}}{2}$.

For any $L>0$, we have by estimate (B.6)

$\displaystyle P_{0,t_{0}+1,h}(w_{0},H\setminus B_{L}^{H_{2}})\leq
L^{-2}\mathbf{E}_{w_{0}}\|\Phi_{0,t_{0}+1,h}\|_{2}^{2}\leq CL^{-2},$

where the constant $C=C(R,t_{0})$ does not depend on $h$. There is
$L_{0}=L_{0}(R)$ such that for all $L\geq L_{0}$,

$\sup_{(w_{0},h)\in B_{R}^{H}\times\mathbb{T}}P_{0,t_{0}+1,h}(w_{0},H\setminus
B_{L}^{H_{2}})\leq p_{0}/4.$

Hence by (C.22), we deduce

$\displaystyle P_{0,t_{0}+1,h}(w_{0},B^{H,B_{L}^{H_{2}}}_{r}(w))$
$\displaystyle=P_{0,t_{0}+1,h}(w_{0},B^{H,H_{2}}_{r}(w))-P_{0,t_{0}+1,h}(w_{0},B^{H,H_{2}\setminus
B_{L}^{H_{2}}}_{r}(w))$ $\displaystyle\geq p_{0}/2-p_{0}/4=p_{0}/4,\quad\text{
for all }(w_{0},h,w)\in B_{R}^{H}\times\mathbb{T}\times B_{\rho}^{H_{2}}.$

Inequality (C.18) then follows. The proof is complete. ∎

###  C.3. Uniform Feller Property

In this subsection we prove the uniform Feller property. We follow the
arguments presented in [31] and use the cocycle property of the inhomogeneous
Feynman-Kac family.

######  Proposition C.4 (Uniform Feller Property).

Assume $A_{\infty}=H,f\in C_{b}(\mathbb{R},H_{2})$ in (2.2). For any $V\in
C_{b}^{1,0}(H\times\mathbb{T})$ and $\phi\in C_{b}^{1}(H\times\mathbb{T})$
there is an $R_{0}=R_{0}(V)>0$ such that for any $R>0$ the family

$\left\\{\frac{P_{0,t,h}^{V}\phi(\cdot,\beta_{t}h)}{\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}},t\geq
0,h\in\mathbb{T}\right\\}$

is uniformly Lipschitz on $B_{R}^{H}$ with a uniform bound on the Lipschitz
constants depending on

$R,\|\phi\|_{\infty},\|V\|_{\infty},\sup_{h\in\mathbb{T}}\|\nabla\phi(\cdot,h)\|_{\infty},\sup_{h\in\mathbb{T}}\|\nabla
V(\cdot,h)\|_{\infty}.$

In particular, the family is uniformly equicontinuous on
$(B_{R}^{H_{2}},\|\cdot\|)$ for any $R>0$.

Here $\nabla$ is the gradient in the $H$ component. For convenience we will
denote

$\|\nabla V\|_{\infty}:=\sup_{h\in\mathbb{T}}\|\nabla
V(\cdot,h)\|_{\infty},\,\text{ for }V\in C_{b}^{1,0}(H\times\mathbb{T}).$

Proposition C.4 is a direct consequence of the following estimate.

######  Proposition C.5.

Assume $A_{\infty}=H,f\in C_{b}(\mathbb{R},H_{2})$ in (2.2). Then for any
$V\in C_{b}^{1,0}(H\times\mathbb{T}),\phi\in C_{b}^{1}(H\times\mathbb{T})$ and
$c,\eta>0$, there are $R_{0}=R_{0}(V)$ and a constant $C$ independent of $h$
such that

(C.23) $\displaystyle\|\nabla\mathcal{P}^{V}_{t}\phi(w,h)\|\leq
C\exp(\eta\|w\|^{2})\|P_{0,t,h}^{V}\mathbb{1}\|_{R_{0}}\left(\|\phi\|_{\infty}+e^{-ct}\|\nabla\phi\|_{\infty}\right),$

for any $(w,h)\in H\times\mathbb{T}$ and $t\geq 0$.

This estimate is proved through Malliavin calculus. In the time homogeneous
case this type of gradient estimate was proved in [31] recently. Since the
current work is in a time inhomogeneous setting, we include a detailed proof
below, which basically follows from the scheme in [31]. The time inhomogeneity
is overcome by the translation identity (2.6) and the cocycle property (2.7).
The proof of Proposition C.5 is giving in subsection C.3.2, after we give some
basic facts about Malliavin calculus in subsection C.3.1.

####  C.3.1. Malliavin calculus preliminaries

Recall that $\Phi_{0,t,h}(w_{0},W)=\Phi_{0,t,h}(w_{0},W\mathbb{1}_{[0,t]})$ is
the solution of the stochastic Navier-Stokes equation at time $t$, where $W$
is the Wiener process. The Malliavin derivative of this random variable in the
direction of $v\in\mathrm{L}^{2}([0,\infty),\mathbb{R}^{d})$ is

$\displaystyle D^{v}\Phi_{0,t,h}(w_{0}):=\lim_{\varepsilon\rightarrow
0}\frac{\mathrm{\Phi}_{0,t,h}(w_{0},W+\varepsilon
V)-\mathrm{\Phi}_{0,t,h}(w_{0},W)}{\varepsilon},$

where $V(t)=\int_{0}^{t}v(r)dr$. The above limit exists almost surely and one
has

$\displaystyle D^{v}\Phi_{0,t,h}(w_{0})=\int_{0}^{t}J_{r,t,h}Gv(r)dr,$

where $J_{r,t,h}$ is the Jacobian that solves the linearized equation

$\displaystyle\partial_{t}J_{r,t,h}\xi=\nu\mathrm{\Delta}J_{r,t,h}\xi+\widetilde{B}(\Phi_{0,t,h},J_{r,t,h}\xi),\quad
t>r,\quad J_{r,r,h}\xi=\xi,$

for $\xi\in H$. See [21, 32] and references therein for more about Malliavin
calculus.

For any $0\leq\tau\leq t$, the Malliavin operator is defined as
$M_{\tau,t,h}:=A_{\tau,t,h}A_{\tau,t,h}^{*}$, where the random operator
$A_{\tau,t,h}:\mathrm{L}^{2}([0,\infty),\mathbb{R}^{d})\to H$ is given by

$\displaystyle
A_{\tau,t,h}v:=\int_{\tau}^{t}J_{r,t,h}Gv(r)dr,\,v\in\mathrm{L}^{2}([0,\infty),\mathbb{R}^{d}),$

and $A_{\tau,t,h}^{*}$ is its adjoint by the relation

$\left\langle A_{\tau,t,h}v,u\right\rangle_{H}=\left\langle
v,A_{\tau,t,h}^{*}u\right\rangle_{\mathrm{L}^{2}([0,\infty),\mathbb{R}^{d})},\,u\in
H,v\in\mathrm{L}^{2}([0,\infty),\mathbb{R}^{d}).$

It is known that the Tikhonov regularization
$\widetilde{M}_{\tau,t,h}:=M_{\tau,t,h}+\beta$, is invertible for any
$\beta>0$.

Denote $\Xi_{t}^{V}=\exp\left(\int_{0}^{t}V(H_{r})dr\right)$, and
$v_{0,t}=v\mathbb{1}_{[0,t]}$, and set

$\mathfrak{R}_{t,h}=J_{0,t,h}\xi-A_{0,t,h}v_{0,t},$

which is the error between the infinitesimal variation on the Wiener path $W$
by $v$ and the variation on the initial condition by $\xi$ of the solution
process. Applying the integration by parts formula, chain rule and product
rule for the Malliavin derivative [32], for any $w,\xi\in H$ with $\|\xi\|=1$,
and $\mathbf{E}=\mathbf{E}_{(w,h)}$, we have

(C.24)
$\displaystyle\begin{split}\Big{\langle}\nabla&\mathcal{P}_{t}^{V}\phi(w,h),\xi\big{\rangle}=\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}\nabla
V(H_{r})J_{0,r,h}\xi dr+\Xi_{t}^{V}\nabla\phi(H_{t})J_{0,t,h}\xi\right]\\\
&\begin{aligned} &=\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}\nabla
V(H_{r})D^{v}\Phi_{0,r,h}dr+\Xi_{t}^{V}\nabla\phi(H_{t})D^{v}\Phi_{0,t,h}\right]\\\
&\qquad+\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}\nabla
V(H_{r})\mathfrak{R}_{r,h}dr+\Xi_{t}^{V}\nabla\phi(H_{t})\mathfrak{R}_{t,h}\right]\end{aligned}\\\
&\begin{aligned}
&=\mathbf{E}\left[D^{v}\left(\Xi_{t}^{V}\phi(H_{t})\right)\right]+\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}\nabla
V(H_{r})\mathfrak{R}_{r,h}dr\right]+\mathbf{E}\left[\Xi_{t}^{V}\nabla\phi(H_{t})\mathfrak{R}_{t,h}\right]\end{aligned}\\\
&\begin{aligned}
&=\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}v(r)dW(r)\right]+\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}\nabla
V(H_{r})\mathfrak{R}_{r,h}dr\right]+\mathbf{E}\left[\Xi_{t}^{V}\nabla\phi(H_{t})\mathfrak{R}_{t,h}\right]\end{aligned}\\\
&:=I_{1}+I_{2}+I_{3},\end{split}$

where we used the fact that $D^{v}\Phi_{0,t,h}=A_{0,t}v_{0,t}$. The gradient
inequality (C.23) will be established if one can choose an appropriate control
$v$ to have desired bounds on $I_{j},j=1,2,3$. Here we choose essentially the
same control $v$ as in [21],

$\displaystyle
v(r)=\left\\{\begin{array}[]{ll}A_{2n}^{*}\widetilde{M}_{2n}^{-1}J_{2n}\mathfrak{R}_{2n,h}&\text{
for }r\in[2n,2n+1),n\geq 0,\\\ 0&\text{ for }r\in[2n+1,2n+2),n\geq
0,\end{array}\right.$

where $J_{n}=J_{n,n+1,h}$, $A_{n}=A_{n,n+1,h}$, $M_{n}=A_{n}A_{n}^{*}$,
$\widetilde{M}_{n}=\beta+M_{n}$. Note that the control is not always adapted,
hence the integral in $I_{1}$ is understood as a Skorokhod integral.

The following result follows from the framework developed in [21]. We refer
interested readers to [27] for a verification in the current setting.

######  Proposition C.6.

For any $\eta>0$ and $a>0$ there are constants $\beta>0$ and $C=C(\eta,a)$, so
that

(C.25) $\displaystyle\mathbf{E}\|\mathfrak{R}_{t}\|^{2}\leq
C\exp(\eta\|w\|^{2})e^{-2at},\,\forall t\geq 0,$ (C.26)
$\displaystyle\mathbf{E}\left|\int_{2n}^{t}v(r)dW(r)\right|^{2}\leq
C\exp(\eta\|w\|^{2})e^{-2an},\,\forall t\in[2n,2n+2],$

for all $n\geq 0$, $h\in\mathbb{T},w\in H$.

####  C.3.2. Proof of Proposition C.5

We now prove Proposition C.5 by combining inequality (C.24), Proposition C.6
and the growth estimates from Proposition C.1, through giving desired
estimates on $I_{j},j=1,2,3$ in inequality (C.24). Let $R_{0}$ and $q$ be the
numbers as in Proposition C.1 and assume $V\geq 0$ without loss of generality.
The proof is divided into three steps.

Step 1: Estimate on $I_{1}$. The idea is to use the decay in (C.26) to control
the growth resulted by the potential $V$ on each single interval. Let $N_{t}$
be the largest integer such that $2N_{t}\leq t$. We can rewrite $I_{1}$ as
follows,

$\displaystyle
I_{1}=\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}v(r)dW(r)\right]$
$\displaystyle=\sum_{n=1}^{N_{t}}\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{2n-2}^{2n}v(r)dW(r)\right]+\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{2N_{t}}^{t}v(r)dW(r)\right]$
$\displaystyle:=I_{1,1}+I_{1,2}.$

By the construction of $v$, we know that for each $1\leq n\leq N_{t}$, the
integral $\int_{2n-2}^{2n}v(r)dW(r)$ is $\mathcal{F}_{2n}$ measurable.
Therefore by the Markov property, we have

(C.27)
$\displaystyle\begin{split}\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{2n-2}^{2n}v(r)dW(r)\right]&=\mathbf{E}\left[\int_{2n-2}^{2n}v(r)dW(r)\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})|\mathcal{F}_{2n}\right]\right]\\\
&=\mathbf{E}\left[\Xi_{2n}^{V}\int_{2n-2}^{2n}v(r)dW(r)\mathcal{P}_{t-2n}^{V}\phi(\mathcal{H}_{2n})\right].\end{split}$

By Proposition C.1,

$\displaystyle\left|\mathcal{P}_{t-2n}^{V}\phi(w,h)\right|=\left|P^{V}_{0,t-2n,h}\phi(\cdot,\beta_{t-2n}h)(w)\right|$
$\displaystyle\leq\|\phi\|_{\infty}P^{V}_{0,t-2n,h}\mathbb{1}_{H}(w)$
$\displaystyle\leq
C\|\phi\|_{\infty}\mathfrak{m}_{q}(w)\|P_{0,t-2n,h}^{V}\mathbb{1}_{H}\|_{R_{0}}.$

Therefore by $V\geq 0$ and translation identity (2.6),

(C.28)
$\displaystyle\begin{split}\left|\mathcal{P}_{t-2n}^{V}\phi(\mathcal{H}_{2n})\right|\leq
C\|\phi\|_{\infty}\mathfrak{m}_{q}(\Phi_{0,2n,h})\|P_{0,t-2n,\beta_{2n}h}^{V}\mathbb{1}_{H}\|_{R_{0}}&=C\|\phi\|_{\infty}\mathfrak{m}_{q}(\Phi_{0,2n,h})\|P_{2n,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\\\
&\leq
C\|\phi\|_{\infty}\mathfrak{m}_{q}(\Phi_{0,2n,h})\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}.\end{split}$

Combining this with (C.27), estimate (B.6), Hölder’s inequality, and (C.26) we
obtain

$\displaystyle\Bigg{|}\mathbf{E}\Big{[}\Xi_{t}^{V}\phi(H_{t})\int_{2n-2}^{2n}v(r)$
$\displaystyle dW(r)\Big{]}\Bigg{|}\leq
C\|\phi\|_{\infty}e^{2\|V\|_{\infty}n}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\left|\mathbf{E}\left[\int_{2n-2}^{2n}v(r)dW(r)\mathfrak{m}_{q}(\Phi_{0,2n,h})\right]\right|$
$\displaystyle\leq
C\|\phi\|_{\infty}e^{2\|V\|_{\infty}n}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\left(\mathbf{E}\left|\int_{2n-2}^{2n}v(r)dW(r)\right|^{2}\right)^{\frac{1}{2}}\left(\mathbf{E}\mathfrak{m}_{2q}(\Phi_{0,2n,h})\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\|\phi\|_{\infty}e^{(2\|V\|_{\infty}-a)n}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\mathfrak{m}_{\eta}(w).$

As a result,

$\displaystyle\left|I_{1,1}\right|\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\mathfrak{m}_{\eta}(w)\sum_{n=1}^{N_{t}}e^{(2\|V\|_{\infty}-a)n}.$

On the other hand, again by $V\geq 0$ and estimate (C.26), one has

$\displaystyle\left|I_{1,2}\right|$
$\displaystyle=\left|\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{2N_{t}}^{t}v(r)dW(r)\right]\right|$
$\displaystyle\leq\|\phi\|_{\infty}e^{\|V\|_{\infty}t}\mathbf{E}\left|\int_{2N_{t}}^{t}v(r)dW(r)\right|\leq
Ce^{(\|V\|_{\infty}-a)n}.$

Therefore we obtain

(C.29)
$\displaystyle\left|I_{1}\right|\leq\left|I_{1,1}\right|+\left|I_{1,2}\right|\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\mathfrak{m}_{\eta}(w)\sum_{n=1}^{N_{t}}e^{(2\|V\|_{\infty}-a)n}.$

The proof of Step 1 is complete.

Step 2: Estimate on $I_{2}$. The calculation of $I_{2}$ is similar to that of
$I_{1}$. The difference is that now we will use (C.25) and $\mathcal{F}_{r*}$
measurability of $\mathfrak{R}_{r,h}$, where $r^{*}=r$ if $\lceil r\rceil$,
the smallest integer greater than or equal to $r$, is even, and $r^{*}=\lceil
r\rceil$ if $\lceil r\rceil$ is odd. We first rewrite

$\displaystyle I_{2}$
$\displaystyle=\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{t}\nabla
V(H_{r})\mathfrak{R}_{r,h}dr\right]$
$\displaystyle=\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{0}^{N_{t}}\nabla
V(H_{r})\mathfrak{R}_{r,h}dr\right]+\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\int_{N_{t}}^{t}\nabla
V(H_{r})\mathfrak{R}_{r,h}dr\right]$ $\displaystyle:=I_{2,1}+I_{2,2},$

where $N_{t}$ is the largest even integer such that $N_{t}\leq t$. By the
Markov property, we have

$\displaystyle I_{2,1}$
$\displaystyle=\mathbf{E}\int_{0}^{N_{t}}\mathbf{E}\left[\Xi_{t}^{V}\phi(H_{t})\nabla
V(H_{r})\mathfrak{R}_{r,h}|\mathcal{F}_{r^{*}}\right]dr$
$\displaystyle=\mathbf{E}\int_{0}^{N_{t}}\Xi_{r^{*}}^{V}\nabla
V(H_{r})\mathfrak{R}_{r,h}\mathcal{P}^{V}_{t-r^{*}}\phi(\mathcal{H}_{r^{*}})dr$

Similar to the arguments in deriving (C.28), by Proposition C.1, $V\geq 0$ and
translation identity (2.6), and boundedness of $\phi$, we have

$\displaystyle\left|\mathcal{P}^{V}_{t-r^{*}}\phi(\mathcal{H}_{r^{*}})\right|\leq
C\|\phi\|_{\infty}\mathfrak{m}_{q}(\Phi_{0,r^{*},h})\|P_{0,t-r^{*},\beta_{r^{*}}h}^{V}\mathbb{1}_{H}\|_{R_{0}}$
$\displaystyle=C\|\phi\|_{\infty}\mathfrak{m}_{q}(\Phi_{0,r^{*},h})\|P_{r^{*},t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}$
$\displaystyle\leq
C\|\phi\|_{\infty}\mathfrak{m}_{q}(\Phi_{0,r^{*},h})\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}.$

Hence by estimate (B.6), Hölder’s inequality and (C.25), one has

$\displaystyle\left|I_{2,1}\right|$ $\displaystyle\leq
C\|\phi\|_{\infty}\|\nabla
V\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\int_{0}^{N_{t}}e^{\|V\|_{\infty}r^{*}}\mathbf{E}\left[\|\mathfrak{R}_{r,h}\|\mathfrak{m}_{q}(\Phi_{0,r^{*},h})\right]dr$
$\displaystyle\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\int_{0}^{N_{t}}e^{\|V\|_{\infty}r^{*}}\left(\mathbf{E}\|\mathfrak{R}_{r,h}\|^{2}\right)^{\frac{1}{2}}\left(\mathbf{E}\mathfrak{m}_{2q}(\Phi_{0,r^{*},h})\right)^{\frac{1}{2}}dr$
$\displaystyle\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\exp(\eta\|w\|^{2})\int_{0}^{N_{t}}e^{\|V\|_{\infty}r^{*}-ar}dr$
$\displaystyle\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\exp(\eta\|w\|^{2})\int_{0}^{N_{t}}e^{(2\|V\|_{\infty}-a)r}dr,$

since $r^{*}\leq r+1$.

Now we look at $I_{2,2}$. Noting that by (C.25) and
$\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\geq 1$, we have

$\displaystyle|I_{2,2}|$ $\displaystyle\leq
C\|\phi\|_{\infty}e^{\|V\|_{\infty}t}\int_{N_{t}}^{t}\mathbf{E}\|\mathfrak{R}_{r,h}\|dr$
$\displaystyle\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\exp(\eta\|w\|^{2})e^{\|V\|_{\infty}t}\int_{N_{t}}^{\infty}e^{-ar}dr$
$\displaystyle\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\exp(\eta\|w\|^{2})e^{(2\|V\|_{\infty}-a)t}$

Combining estimates for $I_{2,1}$ and $I_{2,2}$ we obtain

(C.30) $\displaystyle\left|I_{2}\right|\leq
C\|\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\exp(\eta\|w\|^{2})\left(\int_{0}^{N_{t}}e^{(2\|V\|_{\infty}-a)r}dr+e^{(2\|V\|_{\infty}-a)t}\right).$

Step 3: Estimate on $I_{3}$. The calculation for $I_{3}$ follows directly from
(C.25). Indeed,

(C.31)
$\displaystyle\begin{split}\left|I_{3}\right|&=\left|\mathbf{E}\left[\Xi_{t}^{V}\nabla\phi(H_{t})\mathfrak{R}_{t,h}\right]\right|\leq
C\|\nabla\phi\|_{\infty}e^{\|V\|_{\infty}t}\mathbf{E}\|\mathfrak{R}_{t,h}\|\\\
&\leq
C\|\nabla\phi\|_{\infty}\|P_{0,t,h}^{V}\mathbb{1}_{H}\|_{R_{0}}\exp(\eta\|w\|^{2})e^{(2\|V\|_{\infty}-a)t}\end{split}$

Now combining (C.24) and (C.29)-(C.31), for any $c>0$, we can choose $a>0$
such that $2\|V\|_{\infty}-a=-c$, then we obtain the desired gradient estimate
(C.23)

$\displaystyle\|\nabla\mathcal{P}^{V}_{t}\phi(w,h)\|\leq
C\exp(\eta\|w\|^{2})\|P_{0,t,h}^{V}\mathbb{1}\|_{R_{0}}\left(\|\phi\|_{\infty}+e^{-ct}\|\nabla\phi\|_{\infty}\right),$

where $C=C(c,\eta,\|V\|_{\infty},\|\nabla V\|_{\infty})>0$ is independent of
$h\in\mathbb{T}$.

The proof of Proposition C.5 is complete. ∎

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