C^{\overline{\gamma}_{0}}_{2\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$.
Hence for $\eta\in(0,\eta_{0}/2]$,

$\chi^{2}\in C^{\overline{\gamma}_{0}}_{4\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{4\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n}).$

Since $\overline{\gamma}_{0}<\gamma$, it follows from Proposition 4.5 that for
$\eta\in(0,2^{-3}\eta_{0}]$, one has

$P_{1}\chi\in C^{\overline{\gamma}_{0}}_{4\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{4\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n}),$

and for $t\in[0,1]$,

$\widetilde{\phi}P_{1-t}\chi,\chi P_{1}\chi,P_{1}\chi^{2}\in
C^{\overline{\gamma}_{0}}_{2^{3}\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{2^{3}\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n}).$

It then follows from (4.8) and (4.9) that for $\eta\in(0,2^{-4}\eta_{0}]$,

$\int_{0}^{1}P_{t}(\widetilde{\phi}P_{1-t}\chi)(w,h)dt\in
C^{\overline{\gamma}_{0}}_{2^{4}\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{2^{4}\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n}).$

In a similar way, one can deduce that the remaining two integrals in (4.21)
also belongs to the same function space. This shows that $Y\in
C^{\overline{\gamma}_{0}}_{2^{4}\eta,H}(H\times\mathbb{T}^{n})\cap
C^{\overline{\gamma}_{0}}_{2^{4}\eta,\mathbb{T}^{n}}(H\times\mathbb{T}^{n})$.

Note that

$\displaystyle|F(h_{1})-F(h_{2})|$
$\displaystyle\leq\left|\langle\Gamma_{h_{1}},Y(\cdot,h_{1})\rangle-\langle\Gamma_{h_{2}},Y(\cdot,h_{1})\rangle\right|+\left|\langle\Gamma_{h_{2}},Y(\cdot,h_{1})\rangle-\langle\Gamma_{h_{2}},Y(\cdot,h_{2})\rangle\right|$
$\displaystyle:=I_{1}+I_{2}.$

And using the same functions $\chi_{R},\overline{\chi}_{R}$ as in the proof of
Theorem 4.6, together with the Hölder continuity of $\Gamma_{h}$ and the fact
(4.13), we have for $\eta\in(0,2^{-5}\eta_{0}]$,

$\displaystyle I_{1}$
$\displaystyle\leq\left|\langle\Gamma_{h_{1}}-\Gamma_{h_{2}},(\chi_{R}Y)(\cdot,h_{1})\rangle\right|+\left|\langle\Gamma_{h_{1}}-\Gamma_{h_{2}},(\overline{\chi}_{R}Y)(\cdot,h_{1})\rangle\right|$
$\displaystyle\leq C\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,H}e^{2^{6}\eta
R^{2}}(\rho(\Gamma_{h_{1}},\Gamma_{h_{2}}))^{\frac{1}{2-\overline{\gamma}_{0}}}+C\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,H}e^{-2^{4}\eta
R^{2}}$ $\displaystyle\leq
C\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,H}\left(e^{2^{6}\eta
R^{2}}|h_{1}-h_{2}|^{\frac{\gamma}{2-\overline{\gamma}_{0}}}+e^{-2^{4}\eta
R^{2}}\right),$

where we used the uniform integrability
$\int_{H}e^{2^{5}\eta\|w\|^{2}}\Gamma_{h}(dw)\leq C$ in the second inequality,
which is a consequence of Theorem 3.7 by taking $\kappa=2^{4}$. It then
follows from Lemma 3.8 that

$I_{1}\leq
C\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,H}|h_{1}-h_{2}|^{\frac{\gamma}{5(2-\overline{\gamma}_{0})}}.$

Also note that

$\displaystyle I_{2}$
$\displaystyle\leq\langle\Gamma_{h_{2}},\left|Y(\cdot,h_{1})-Y(\cdot,h_{2})\right|\rangle$
$\displaystyle\leq\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,\mathbb{T}^{n}}|h_{1}-h_{2}|^{\overline{\gamma}_{0}}\int_{H}e^{2^{4}\eta\|w\|^{2}}\Gamma_{h_{2}}(dw)\leq
C\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,\mathbb{T}^{n}}|h_{1}-h_{2}|^{\overline{\gamma}_{0}}.$

Since $\overline{\gamma}_{0}\leq\frac{\gamma}{5(2-\overline{\gamma}_{0})}$, we
deduce that

$|F(h_{1})-F(h_{2})|\leq
C\left(\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,\mathbb{T}^{n}}+\|Y\|_{\overline{\gamma}_{0},2^{4}\eta,H}\right)|h_{1}-h_{2}|^{\overline{\gamma}_{0}},\quad\forall
h_{1},h_{2}\in\mathbb{T}^{n}.$

Hence $F\in C^{\overline{\gamma}_{0}}(\mathbb{T}^{n},\mathbb{R})$.

Equation (4.20) follows from the invariance property of the invariant measure
$\Gamma_{h}(dw)\lambda(dh)$ and the decomposition (4.21). Indeed, letting
$m(dwdh)=\Gamma_{h}(dw)\lambda(dh)$ and
$\chi_{t}=\int_{0}^{t}P_{r}\widetilde{\phi}dr$, then by the invariance of $m$
under $P_{t}$, one has

$\displaystyle\int_{0}^{1}\int_{0}^{t}\left\langle
m,P_{\tau}\left(\widetilde{\phi}P_{t-\tau}\widetilde{\phi}\right)\right\rangle
d\tau dt=\int_{0}^{1}\int_{0}^{t}\left\langle
m,\widetilde{\phi}P_{\tau}\widetilde{\phi}\right\rangle d\tau
dt=\int_{0}^{1}\left\langle m,\widetilde{\phi}\chi_{t}\right\rangle dt$
$\displaystyle\int_{0}^{1}\left\langle
m,P_{t}(\widetilde{\phi}P_{1-t}\chi)\right\rangle dt=\int_{0}^{1}\left\langle
m,\widetilde{\phi}P_{t}\chi\right\rangle dt=\left\langle
m,\widetilde{\phi}\chi\right\rangle-\int_{0}^{1}\left\langle
m,\widetilde{\phi}\chi_{t}\right\rangle dt,$

where we used the fact that $P_{t}\chi=\chi-\chi_{t}$. Hence from (4.21) we
have

$\displaystyle\int_{H\times\mathbb{T}^{n}}Y(w,h)dt\Gamma_{h}(dw)\lambda(dh)=\left\langle
m,Y\right\rangle$ $\displaystyle=2\left\langle
m,\chi^{2}\right\rangle-2\left\langle m,\chi
P_{1}\chi\right\rangle+2\left\langle
m,\widetilde{\phi}\chi\right\rangle-2\int_{0}^{1}\left\langle
m,\widetilde{\phi}\chi_{t}\right\rangle dt-2\left\langle
m,\chi\chi_{1}\right\rangle+2\int_{0}^{1}\left\langle
m,\widetilde{\phi}\chi_{t}\right\rangle dt$ $\displaystyle=2\left\langle
m,\chi^{2}\right\rangle-2\left\langle
m,\chi(\chi-\chi_{1})\right\rangle+2\left\langle
m,\widetilde{\phi}\chi\right\rangle-2\left\langle m,\chi\chi_{1}\right\rangle$
$\displaystyle=2\left\langle m,\widetilde{\phi}\chi\right\rangle=\sigma^{2}$

as in (4.18). The proof is complete. ∎

We now proceed to prove the central limit theorem in Theorem 4.2. This is done
by utilizing the following martingale central limit theorem to the
approximating martingale $M_{N}$ in (4.14). It is notable that the theorem
only requires a weak form of law of large numbers for the martingale
difference and Lindeberg type negligible conditions. Basically these
conditions can be derived from the ergodic properties of the homogenized
process as in Proposition 3.11 and certain moment bounds on the martingale
from Lemma 4.9. Although Theorem 4.2 is a direct consequence of Theorem 4.4
that will be proved in the next subsection, we supply the proof for Theorem
4.2 below since it does not require a deep analysis of the convergence of the
conditioned martingale difference to the asymptotic variance.

######  Theorem 4.14 ([29]).

Assume the martingale $M_{N}$, its quadratic variation $[M]_{N}$ and the
associated martingale difference $Z_{N}=M_{N}-M_{N-1}$ (with $M_{0}=0$)
satisfy the following
1.(The Lindeberg type conditions) For every $\varepsilon>0,$

(4.22)
$\displaystyle\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{j=0}^{N-1}\mathbf{E}\left[Z_{j+1}^{2},\left|Z_{j+1}\right|\geq\varepsilon\sqrt{N}\right]=0,$
(4.23)
$\displaystyle\lim_{K\rightarrow\infty}\limsup_{\ell\rightarrow\infty}\frac{1}{\ell
K}\sum_{m=1}^{\ell}\sum_{j=(m-1)K}^{mK-1}\mathbf{E}\left[1+Z_{j+1}^{2},\left|M_{j}-M_{(m-1)K}\right|\geq\varepsilon\sqrt{\ell
K}\right]=0.$

2\. (Law of large numbers for the conditioned martingale difference) There
exists $\sigma\geq 0$ such that

(4.24)
$\displaystyle\lim_{K\rightarrow\infty}\limsup_{\ell\rightarrow\infty}\frac{1}{\ell}\sum_{m=1}^{\ell}\mathbf{E}\left|\frac{1}{K}\mathbf{E}\left[[M]_{mK}-[M]_{(m-1)K}\,\middle|\,\mathfrak{F}_{(m-1)K}\right]-\sigma^{2}\right|=0,$

with uniformly square integrable condition $\displaystyle\sup_{n\geq
1}\mathbf{E}Z_{n}^{2}<+\infty$.
Then one has

$\lim_{N\rightarrow\infty}\frac{\mathbf{E}[M]_{N}}{N}=\sigma^{2},$

and

$\lim_{N\rightarrow\infty}\mathbf{E}e^{i\theta
M_{N}/\sqrt{N}}=e^{-\sigma^{2}\theta^{2}/2},\quad\forall\theta\in\mathbb{R}.$

Before proving Theorem 4.2 by verifying the conditions of the above Theorem
4.14, we recall a useful lemma.

######  Lemma 4.15.

[29] If $\\{\mu_{N}\\}_{N\geq 1}\subset\mathcal{P}(H\times\mathbb{T}^{n})$
converges to $\mu$ weakly, $\\{F_{N}\\}_{N\geq 1}\subset
C(H\times\mathbb{T}^{n})$ converges to $0$ uniformly on compact sets and there
is $\eta>0$ such that
$\limsup_{N\rightarrow\infty}\langle\mu_{N},|F_{N}|^{1+\eta}\rangle<\infty$,
then $\lim_{N\rightarrow\infty}\langle\mu_{N},F_{N}\rangle=0$.

###### Proof of Theorem 4.2.

We first prove the The Lindeberg type conditions. From the Markov property of
the homogenized process, the left hand side of (4.22) can be rewritten as

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\mathbf{E}\left[Z_{j}^{2};|Z_{j}|\geq\varepsilon\sqrt{N}\right]=\frac{1}{N}\sum_{j=1}^{N}\left\langle
P_{j-1}^{*}\delta_{(w,h)},G_{N}\right\rangle,$

where
$G_{N}(u,g)=\mathbf{E}_{(u,g)}[M_{1}^{2};|M_{1}|\geq\varepsilon\sqrt{N}]$ for
$(u,g)\in H\times\mathbb{T}^{n}$. By the Markov inequality and Lemma 4.9, one
has for $\eta\in(0,2^{-3}\eta_{0}]$,

$\displaystyle
G_{N}\leq\left(\mathbf{E}_{(u,g)}|M_{1}|^{4}\right)^{\frac{1}{2}}\mathbf{P}\left(|M_{1}|\geq\varepsilon\sqrt{N}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\mathbf{E}_{(u,g)}|M_{1}|^{4}\right)^{\frac{1}{2}}\left(\frac{\mathbf{E}_{(u,g)}|M_{1}|^{4}}{\varepsilon^{4}N^{2}}\right)^{\frac{1}{2}}\leq
C\varepsilon^{-2}N^{-1}e^{2^{4}\eta\|w\|^{2}}.$

Hence $G_{N}\rightarrow 0$ uniformly on any compact set. Also by Lemma 4.9, it
follows that

$\displaystyle\frac{1}{N}\sum_{j=1}^{N}\left\langle
P_{j-1}^{*}\delta_{(w,h)},G_{N}^{2}\right\rangle\leq\frac{1}{N}\sum_{j=1}^{N}P_{j-1}\mathbf{E}_{(w,h)}|M_{1}|^{4}\leq
Ce^{2^{4}\eta\|w\|^{2}},$

where $C$ is independent of $N$. Therefore

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{j=1}^{N}\left\langle
P_{j-1}^{*}\delta_{(w,h)},G_{N}^{2}\right\rangle<\infty.$

The first Lindeberg condition (4.22) then follows from Proposition 3.11 and
Lemma 4.15.

Again by the Markov property we can rewrite the left hand side formula in the
limit of (4.23) as

$\frac{1}{\ell
K}\sum_{m=1}^{\ell}\sum_{j=(m-1)K}^{mK-1}\mathbf{E}\left[1+Z_{j+1}^{2},\left|M_{j}-M_{(m-1)K}\right|\geq\varepsilon\sqrt{\ell
K}\right]=\frac{1}{K}\sum_{j=0}^{K-1}\left\langle\frac{1}{\ell}\sum_{m=1}^{\ell}P^{*}_{(m-1)K}\delta_{(w,h)},F_{\ell,j}\right\rangle,$

where
$F_{\ell,j}(u,g)=\mathbf{E}_{(u,g)}\left[1+Z_{j+1}^{2},\left|M_{j}\right|\geq\varepsilon\sqrt{\ell
K}\right].$ It follows from the Markov inequality that

$\displaystyle F_{\ell,j}(u,g)$
$\displaystyle\leq\left(\mathbf{E}_{(u,g)}\left(1+Z_{j+1}^{2}\right)^{2}\right)^{\frac{1}{2}}\mathbf{P}\left(|M_{j}|(u,g)\geq\varepsilon\sqrt{\ell
K}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\mathbf{E}_{(u,g)}\left(1+Z_{j+1}^{2}\right)^{2}\right)^{\frac{1}{2}}\frac{\left(\mathbf{E}_{(u,g)}|M_{j}|^{4}\right)^{\frac{1}{2}}}{\left(\varepsilon\sqrt{\ell
K}\right)^{2}}$ $\displaystyle\leq
C(\varepsilon)\frac{1+\mathbf{E}_{(u,g)}|M_{j+1}|^{4}+\mathbf{E}_{(u,g)}|M_{j}|^{4}}{\ell
K}$

In view of Lemma 4.9, we know that for any $R>0$, and $0\leq j\leq K$, there
is a constant $C$ independent of $\ell$ such that

$\sup_{(u,g)\in B_{R}(0)\times\mathbb{T}^{n}}F_{\ell,j}(u,g)\leq\frac{C}{(\ell
K)^{1/2}}.$

Hence $F_{\ell,j}\rightarrow 0$ as $\ell\rightarrow\infty$ uniformly on
bounded (in particular compact) sets. Again by Lemma 4.9,

$\displaystyle\left\langle\frac{1}{\ell}\sum_{m=1}^{\ell}P^{*}_{(m-1)K}\delta_{(w,h)},F_{\ell,j}^{2}\right\rangle\leq
C\left(1+\frac{1}{\ell}\sum_{m=1}^{\ell}P_{(m-1)K}\mathbf{E}_{(w,h)}\left(|M_{j}|^{4}+|M_{j+1}|^{4}\right)\right)\leq
C(K,w).$

Therefore

$\limsup_{\ell\rightarrow\infty}\left\langle\frac{1}{\ell}\sum_{m=1}^{\ell}P^{*}_{(m-1)K}\delta_{(w,h)},F_{\ell,j}^{2}\right\rangle<\infty.$

It then follows from Proposition 3.11 and Lemma 4.15 that

$\lim_{\ell\rightarrow\infty}\left\langle\frac{1}{\ell}\sum_{m=1}^{\ell}P^{*}_{(m-1)K}\delta_{(w,h)},F_{\ell,j}\right\rangle=0,\quad
j=0,\cdots,K-1,$

which completes the proof of (4.23).

Now we proceed to show the law of large numbers for the conditioned martingale
difference as in (4.24). The finiteness of $\displaystyle\sup_{n\geq
1}\mathbf{E}_{(w,h)}Z_{n}^{2}$ follows from Lemma 4.9. By the Markov property,
we have

(4.25)
$\displaystyle\frac{1}{\ell}\sum_{m=1}^{\ell}\mathbf{E}\Big{|}\frac{1}{K}\mathbf{E}\left[[M]_{mK}-[M]_{(m-1)K}|\mathcal{F}_{(m-1)K}\right]-\sigma^{2}\Big{|}=\frac{1}{\ell}\sum_{m=1}^{\ell}\Big{\langle}P^{*}_{(m-1)K}\delta_{(w,h)},|H_{K}|\Big{\rangle},$

where

$\displaystyle
H_{K}(u,g)=\mathbf{E}_{(u,g)}\left[\frac{1}{K}[M]_{K}-\sigma^{2}\right]=\mathbf{E}_{(u,g)}\left[\frac{1}{K}M_{K}^{2}-\sigma^{2}\right]=\frac{1}{K}\sum_{j=0}^{K-1}P_{j}Y_{0}(u,g),$

with
$Y_{0}(u,g)=\mathbf{E}_{(u,g)}[M]_{1}-\sigma^{2}=\mathbf{E}_{(u,g)}M_{1}^{2}-\sigma^{2}$.
In view of the decomposition (4.21) for $\mathbf{E}_{(u,g)}M_{1}^{2}$ and
Proposition 4.5, together with Proposition 4.7, we see that for
$\eta\in(0,2^{-4}\eta_{0}]$, $Y_{0}\in
C^{\overline{\gamma}_{0}}_{2^{4}\eta,H}(H\times\mathbb{T}^{n})$ where
$\overline{\gamma}_{0}$ is as in Proposition 4.7. Note that we do not require
the Hölder continuity of $\Psi$ here. Also by Lemma 4.9, we have

$\displaystyle\frac{1}{\ell}\sum_{m=1}^{\ell}\Big{\langle}P^{*}_{(m-1)K}\delta_{(w,h)},|H_{K}|^{2}\Big{\rangle}\leq
C(\sigma,K)\left(1+\frac{1}{\ell}\sum_{m=1}^{\ell}P_{(m-1)K}\mathbf{E}_{(w,h)}M_{K}^{4}\right)$
$\displaystyle\leq C(\sigma,K,w),$

which is independent of $\ell$. Hence

$\limsup_{\ell\rightarrow\infty}\frac{1}{\ell}\sum_{m=1}^{\ell}\Big{\langle}P^{*}_{(m-1)K}\delta_{(w,h)},|H_{K}|^{2}\Big{\rangle}<\infty.$

Now since
$\displaystyle\frac{1}{\ell}\sum_{m=1}^{\ell}P^{*}_{(m-1)K}\delta_{(w,h)}$
converges weakly to $\Gamma_{g}(du)\lambda(dg)$ by Proposition 3.11, it
follows that

$\lim_{\ell\rightarrow\infty}\frac{1}{\ell}\sum_{m=1}^{\ell}\Big{\langle}P^{*}_{(m-1)K}\delta_{(w,h)},|H_{K}|\Big{\rangle}=\int_{H\times\mathbb{T}^{n}}|H_{K}(u,g)|\Gamma_{g}(du)\lambda(dg).$

Since the homogenized process is unique ergodic with invariant measure
$\Gamma_{g}(du)\lambda(dg)$, it follows from the Birkhoff ergodic theorem for
stationary ergodic process, that for an appropriate initial condition $X_{0}$
with law $\Gamma_{g}(du)\lambda(dg)$,

$\displaystyle\lim_{K\rightarrow\infty}\int_{H\times\mathbb{T}^{n}}|H_{K}(u,g)|\Gamma_{g}(du)\lambda(dg)$
$\displaystyle=\lim_{K\rightarrow\infty}\mathbf{E}\left|\frac{1}{K}\sum_{j=0}^{K-1}P_{j}Y_{0}(X_{0})\right|=\lim_{K\rightarrow\infty}\mathbf{E}\left|\frac{1}{K}\sum_{j=0}^{K-1}Y_{0}(X_{j}(X_{0}))\right|$
$\displaystyle=\left|\int_{H\times\mathbb{T}^{n}}Y_{0}(u,g)\Gamma_{g}(du)\lambda(dg)\right|=0,$

provided that
$\int_{H\times\mathbb{T}^{n}}Y_{0}(u,g)\Gamma_{g}(du)\lambda(dg)=0$, i.e.,
$\sigma^{2}=\int_{H\times\mathbb{T}^{n}}\mathbf{E}_{(u,g)}M_{1}^{2}\Gamma_{g}(du)\lambda(dg)$,
which follows from (4.20). This completes the proof of (4.24). Hence the
central limit theorem (4.7) follows from Theorem 4.14, the martingale
approximation (4.14) and Theorem (4.10). ∎

###  4.3. The rate of convergence in the limit theorems

The aim of this subsection is to show the desired rate of convergence in the
limit theorems as in Theorem 4.3 and Theorem 4.4. The proofs are given in
subsection 4.3.1 and subsection 4.3.2 respectively. We begin with a result
that gives a convergence rate for the moments of the time average of the
observations centered by the quasi-periodic invariant measure. It will be
useful when estimating the rate of convergence of the conditioned martingale
difference to the variance.

######  Proposition 4.16.

For any integer $p\geq 1$, $\eta\in(0,\frac{\eta_{0}}{4p}]$, $\gamma\in(0,1]$
and $\phi\in C_{\eta,H}^{\gamma}(H\times\mathbb{T}^{n})$, we have

(4.26)
$\displaystyle\mathbf{E}_{(w,h)}\left|\frac{1}{N}\sum_{k=1}^{N}\Big{(}\phi(X_{k-1})-\big{\langle}\Gamma_{\beta_{k-1}h},\phi(\cdot,\beta_{k-1}h)\big{\rangle}\Big{)}\right|^{2p}\leq
C_{p}e^{4p\eta\|w\|^{2}}\|\phi\|_{\gamma,\eta,H}^{p}N^{-p},$

for all $N\geq 1,(w,h)\in H\times\mathbb{T}^{n}$. The same result also holds
if we replace the summation by integration:

(4.27)
$\displaystyle\mathbf{E}_{(w,h)}\left|\frac{1}{T}\int_{0}^{T}\Big{(}\phi(X_{t})-\big{\langle}\Gamma_{\beta_{t}h},\phi(\cdot,\beta_{t}h)\big{\rangle}\Big{)}dt\right|^{2p}\leq
C_{p}e^{4p\eta\|w\|^{2}}\|\phi\|_{\gamma,\eta,H}^{p}T^{-p},$

for any $T\geq 1$ and $(w,h)\in H\times\mathbb{T}^{n}$.

To show this proposition, we give a lemma first.

######  Lemma 4.17.

For any real numbers $\\{x_{i}\\}_{i\geq 1}$ and any integer $m\geq 1,p\geq
1$, let $\displaystyle S_{m}=\sum_{i=1}^{m}x_{i}$. Then one has

$\displaystyle\left|S_{m}\right|^{2p}=\left|\sum_{i=1}^{m}x_{i}\right|^{2p}=\sum_{i=1}^{m}\sum_{j=i}^{m}f_{2p-2,j}(x_{1},x_{2},\cdots,x_{i-1},x_{i})x_{i}x_{j},$

where

$\displaystyle
f_{2p-2,j}(x_{1},x_{2},\cdots,x_{i-1},x_{i}):=\left\\{\begin{array}[]{cc}\displaystyle\sum_{k=0}^{2p-2}(k+1)S_{i-1}^{k}S_{i}^{2p-2-k},&\quad\text{when}\,j=i,\\\
\displaystyle
2p\sum_{k=0}^{2p-2}S_{i-1}^{k}S_{i}^{2p-2-k},&\quad\text{when}\,j>i.\\\
\end{array}\right.$

###### Proof.

By the multinomial formula, one has for $j=i$,

$\displaystyle f_{2p-2,i}(x_{1},x_{2},\cdots,x_{i-1},x_{i})$
$\displaystyle=\sum_{k_{1}+k_{2}+\cdots+k_{i}=2p-2}\frac{(2p)!}{k_{1}!k_{2}!\cdots
k_{i-1}!(k_{i}+2)!}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{i}^{k_{i}}$
$\displaystyle=\sum_{k_{i}=0}^{2p-2}C_{2p}^{k_{i}+2}S_{i-1}^{2p-2-k_{i}}x_{i}^{k_{i}}$
$\displaystyle=x_{i}^{-2}\left(S_{i}^{2p}-S_{i-1}^{2p}-2pS_{i-1}^{2p-1}x_{i}\right)$
$\displaystyle=x_{i}^{-2}\left(S_{i}\left(S_{i}^{2p-1}-S_{i-1}^{2p-1}\right)-(2p-1)S_{i-1}^{2p-1}x_{i}\right)$
$\displaystyle=\sum_{k=0}^{2p-2}(k+1)S_{i-1}^{k}S_{i}^{2p-2-k}.$

And similarly for $j>i$,

$\displaystyle f_{2p-2,j}(x_{1},x_{2},\cdots,x_{i-1},x_{i})$
$\displaystyle=\sum_{k_{1}+k_{2}+\cdots+k_{i}=2p-2}\frac{(2p)!}{k_{1}!k_{2}!\cdots
k_{i-1}!(k_{i}+1)!}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{i}^{k_{i}}$
$\displaystyle=2p\sum_{k_{i}=0}^{2p-2}C_{2p-1}^{k_{i}+1}S_{i-1}^{2p-2-k_{i}}x_{i}^{k_{i}}$
$\displaystyle=2px_{i}^{-1}\left(S_{i}^{2p-1}-S_{i-1}^{2p-1}\right)$
$\displaystyle=2p\sum_{k=0}^{2p-2}S_{i-1}^{k}S_{i}^{2p-2-k}.$

∎

###### Proof of Proposition 4.16.

Recall that
$\widetilde{\phi}(w,h)=\phi(w,h)-\langle\Gamma_{h},\phi(\cdot,h)\rangle$, then
the summands in inequality (4.26) is $\xi_{k}:=\widetilde{\phi}(X_{k-1})$. Let

$S_{m}=\sum_{i=1}^{m}\xi_{i},\quad s_{N}=\sup_{1\leq m\leq
N}\mathbf{E}_{(w,h)}|S_{m}|^{2p}.$

Let
$g(\xi_{i},\xi_{j})=\xi_{i}\mathbf{E}_{(w,h)}\left[\xi_{j}|\mathcal{F}_{i-1}\right]$
and
$g_{p}(w,h)=\left(\mathbf{E}_{(w,h)}\left|g(\xi_{i},\xi_{j})\right|^{p}\right)^{1/p}$.
It follows from Lemma 4.17 and the Hölder inequality that

$\displaystyle\mathbf{E}_{(w,h)}|S_{m}|^{2p}=\mathbf{E}_{(w,h)}\sum_{i=1}^{m}\sum_{j=i}^{m}f_{2p-2,j}(\xi_{1},\xi_{2},\cdots,\xi_{i-1},\xi_{i})\xi_{i}\xi_{j}$
$\displaystyle=\sum_{i=1}^{m}\sum_{j=i}^{m}\mathbf{E}_{(w,h)}\Big{[}f_{2p-2,j}(\xi_{1},\xi_{2},\cdots,\xi_{i-1},\xi_{i})\xi_{i}\mathbf{E}_{(w,h)}\left[\xi_{j}|\mathcal{F}_{i-1}\right]\Big{]}$
$\displaystyle\leq\sum_{i=1}^{m}\sum_{j=i}^{m}\left(\mathbf{E}_{(w,h)}\left|f_{2p-2,j}(\xi_{1},\xi_{2},\cdots,\xi_{i-1},\xi_{i})\right|^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}}g_{p}(w,h)$
$\displaystyle\leq\sum_{i=1}^{m}\sum_{j=i}^{m}2p\left((2p-1)^{\frac{1}{p-1}}\sum_{k=0}^{2p-2}\mathbf{E}_{(w,h)}\left(|S_{i-1}|^{k}|S_{i}|^{2p-2-k}\right)^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}}g_{p}(w,h)$
$\displaystyle\leq\sum_{i=1}^{m}\sum_{j=i}^{m}2p(2p-1)^{\frac{1}{p}}\left(\sum_{k=0}^{2p-2}\mathbf{E}_{(w,h)}\left(\frac{k}{2p-2}|S_{i-1}|^{2p-2}+\frac{2p-2-k}{2p-2}|S_{i}|^{2p-2}\right)^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}}g_{p}(w,h)$
$\displaystyle\leq\sum_{i=1}^{m}\sum_{j=i}^{m}2p(2p-1)\Big{(}\mathbf{E}_{(w,h)}\left[|S_{i-1}|^{2p}+|S_{i}|^{2p}\right]\Big{)}^{\frac{p-1}{p}}g_{p}(w,h).$

Taking supremum for $1\leq m\leq N$, one has

$\displaystyle s_{N}\leq
4p(2p-1)s_{N}^{\frac{p-1}{p}}\sum_{i=1}^{N}\sum_{j=i}^{N}\left(\mathbf{E}_{(w,h)}\left|g(\xi_{i},\xi_{j})\right|^{p}\right)^{1/p}.$

Hence by letting $C_{p}=(4p(2p-1))^{p}$,

$\displaystyle s_{N}\leq
C_{p}\left(\sum_{i=1}^{N}\sum_{j=i}^{N}\left(\mathbf{E}_{(w,h)}\left|g(\xi_{i},\xi_{j})\right|^{p}\right)^{1/p}\right)^{p}.$

Note that by Theorem 4.6 with $\kappa=2$,

$\displaystyle\mathbf{E}_{(w,h)}\left|g(\xi_{i},\xi_{j})\right|^{p}$
$\displaystyle=\mathbf{E}_{(w,h)}\left|\widetilde{\phi}(X_{i-1})\mathbf{E}_{(w,h)}\left[\widetilde{\phi}(X_{j-1})|\mathcal{F}_{i-1}\right]\right|^{p}$
$\displaystyle\leq\mathbf{E}_{(w,h)}\left|\widetilde{\phi}(X_{i-1})P_{j-i}\widetilde{\phi}(X_{i-1})\right|^{p}$
$\displaystyle\leq
C^{p}\|\phi\|_{\gamma,\eta,H}^{p}e^{-p\Lambda(j-i)}\mathbf{E}_{(w,h)}e^{4p\eta\|w_{0,i-1,h}(w)\|^{2}}$
$\displaystyle\leq
C^{p}\|\phi\|_{\gamma,\eta,H}^{p}e^{-p\Lambda(j-i)}e^{4p\eta\|w\|^{2}},$

for $\eta\in(0,\frac{\eta_{0}}{4p}]$ by estimate (6.1). Therefore

$\displaystyle s_{N}\leq
CC_{p}\|\phi\|_{\gamma,\eta,H}^{p}e^{4p\eta\|w\|^{2}}N^{p},$

where $C>0$ does not depend on $p$. Dividing both sides of the above
inequality by $N^{2p}$ completes the proof of the first estimate (4.26) in
Proposition 4.16.

The second inequality follows from the same argument (see also [34] for the
case in the time homogeneous setting). Let
$\xi(t)=\phi(X_{t})-\big{\langle}\Gamma_{\beta_{t}h},\phi(\cdot,\beta_{t}h)\big{\rangle}$,
$I_{r}=\int_{0}^{r}\xi(t)dt$ and $\displaystyle\mathcal{I}_{T}=\sup_{0\leq
r\leq T}\mathbf{E}_{(w,h)}\left|I_{r}\right|^{2p}$. We first note that

$\displaystyle I_{r}^{2p}$
$\displaystyle=\int_{[0,r]^{2p}}\xi(t_{1})\xi(t_{2})\cdots\xi(t_{2p})dt_{1}dt_{2}\cdots
dt_{2p}$ $\displaystyle=(2p)!\int_{0\leq t_{1}\leq t_{2}\leq\cdots\leq
t_{2p}\leq r}\xi(t_{1})\xi(t_{2})\cdots\xi(t_{2p})dt_{1}dt_{2}\cdots dt_{2p}.$

For $r_{1}\leq r_{2}$, denote
$\varphi(r_{1},r_{2})=\xi(r_{1})\mathbf{E}_{(w,h)}[\xi(r_{2})|\mathcal{F}_{r_{1}}]$.
Then

$\displaystyle\mathbf{E}_{(w,h)}\left|I_{r}\right|^{2p}=(2p)!\mathbf{E}_{(w,h)}\int_{0\leq
t_{1}\leq t_{2}\leq\cdots\leq t_{2p}\leq
r}\xi(t_{1})\xi(t_{2})\cdots\xi(t_{2p-2})\varphi(t_{2p-1},t_{2p})dt_{1}dt_{2}\cdots
dt_{2p}$
$\displaystyle=(2p)!\mathbf{E}_{(w,h)}\left(\int_{0}^{r}\int_{0}^{t_{2p}}\varphi(t_{2p-1},t_{2p})\left(\int_{0\leq
t_{1}\leq\cdots\leq t_{2p-1}}\xi(t_{1})\cdots\xi(t_{2p-2})dt_{1}\cdots
dt_{2p-1}\right)dt_{2p-1}dt_{2p}\right)$ $\displaystyle\leq
2p(2p-1)\int_{0}^{r}\int_{0}^{t_{2p}}\left(\mathbf{E}_{(w,h)}\left|\varphi(t_{2p-1},t_{2p})\right|^{p}\right)^{\frac{1}{p}}\left(\mathbf{E}_{(w,h)}\left|I_{t_{2p-1}}\right|^{2p}\right)^{\frac{p-1}{p}}dt_{2p-1}dt_{2p}.$

Taking the supremum w.r.t $r$ over $[0,T]$, we have

$\displaystyle\mathcal{I}_{T}\leq
2p(2p-1)\left(\mathcal{I}_{T}\right)^{\frac{p-1}{p}}\int_{0}^{T}\int_{0}^{t_{2}}\left(\mathbf{E}_{(w,h)}\left|\varphi(t_{1},t_{2})\right|^{p}\right)^{\frac{1}{p}}dt_{1}dt_{2}.$

Like in the proof of (4.26), one has for $\eta\in(0,\frac{\eta_{0}}{4p})$,

$\left(\mathbf{E}_{(w,h)}\left|\varphi(t_{1},t_{2})\right|^{p}\right)^{\frac{1}{p}}\leq
C\|\phi\|_{\gamma,\eta,H}e^{-\Lambda(t_{2}-t_{1})}e^{4\eta\|w\|^{2}}.$

Therefore

$\displaystyle\mathcal{I}_{T}\leq
C_{p}\|\phi\|_{\gamma,\eta,H}^{p}T^{p}e^{4p\eta\|w\|^{2}}$

and the inequality (4.27) follows by dividing both sides with $T^{2p}$. ∎

####  4.3.1. The rate of convergence in SLLN

The convergence rate of SLLN is a consequence of (4.27), the error estimate in
Lemma 4.10 and the Borel-Cantelli lemma. We now give the details.

###### Proof of Theorem 4.3.

For any $\varepsilon>0$, let
$E_{N}=\left\\{\omega\in\Omega:\left|\frac{1}{N}M_{N}\right|>N^{-(1/2-\varepsilon)}\right\\}$.
From Proposition 4.7, we know that for $\eta\in(0,\eta_{0}/2]$, $\chi\in
C^{\gamma_{0}}_{2\eta,H}(H\times\mathbb{T}^{n})$. Thus $\chi^{2^{p}}\in
C^{\gamma_{0}}_{2^{p+1}\eta,H}(H\times\mathbb{T}^{n})$. Hence from estimate
(4.27) and (6.1) and Markov’s inequality, we have for
$\eta\in(0,2^{-(p+1)}\eta_{0}]$

$\displaystyle\mathbf{P}(E_{N})\leq
N^{2^{p}(-1/2-\varepsilon)}\mathbf{E}_{(w,h)}M_{N}^{2^{p}}$ $\displaystyle\leq
C_{p}N^{2^{p}(-1/2-\varepsilon)}\mathbf{E}_{(w,h)}\left(\chi(X_{N})^{2^{p}}+\chi(w,h)^{2^{p}}+\left(\int_{0}^{N}\widetilde{\phi}(X_{t})dt\right)^{2^{p}}\right)$
$\displaystyle\leq
C_{p}N^{2^{p}(-1/2-\varepsilon)}\left(\|\chi\|_{\gamma_{0},2^{p+1}\eta,H}\mathbf{E}e^{2^{p+1}\eta\|w_{0,N,h}(w)\|^{2}}+\|\chi\|_{\gamma_{0},2^{p+1}\eta,H}e^{2^{p+1}\eta\|w\|^{2}}+e^{2^{p+1}\eta\|w\|^{2}}\|\phi\|_{\gamma,\eta,H}^{2^{p-1}}N^{2^{p-1}}\right)$
$\displaystyle\leq
C_{p}(\|\phi\|_{\gamma,\eta,H})N^{-2^{p}\varepsilon}e^{2^{p+1}\eta\|w\|^{2}}.$

For any $\varepsilon>0$, and every integer $p$ such that $2^{p}\varepsilon>1$,

$\sum_{N=1}^{\infty}\mathbf{P}(E_{N})<\infty.$

By the Borel-Cantelli lemma, there is an almost surely finite random time
$N_{1}(\omega)$, such that for all $N>N_{1}(\omega)$,

$\left|\frac{1}{N}M_{N}\right|\leq N^{-(1/2-\varepsilon)}.$

Note that for $\ell>0$,

$\displaystyle\mathbf{E}N_{1}^{\ell}=\sum_{k=1}^{\infty}\mathbf{P}(N_{1}=k)k^{\ell}$
$\displaystyle\leq\sum_{k=1}^{\infty}\mathbf{P}(E_{k})k^{\ell}$
$\displaystyle\leq\sum_{k=1}^{\infty}C_{p}(\|\phi\|_{\gamma,\eta,H})k^{\ell-2^{p}\varepsilon}e^{2^{p+1}\eta\|w\|^{2}}$
$\displaystyle\leq
C_{p}(\|\phi\|_{\gamma,\eta,H},\ell,\varepsilon)e^{2^{p+1}\eta\|w\|^{2}}$

as long as $\ell<2^{p}\varepsilon-1$. In a similar fashion we can estimate the
moments of the random time $N_{0}(\omega)$ in (4.17). Let $\ell>0$, then for
$\eta\in(0,2^{-p-1}\eta_{0}]$,

$\displaystyle\mathbf{E}N_{0}^{\ell}$
$\displaystyle=\sum_{k=1}^{\infty}\mathbf{P}(N_{0}=k)k^{\ell}$
$\displaystyle\leq\sum_{k=1}^{\infty}\mathbf{P}\left(\sup_{k\leq t\leq
k+1}R_{k,t}>k^{1/4}\right)k^{\ell}\leq\sum_{k=1}^{\infty}\mathbf{E}\left(\sup_{k\leq
t\leq k+1}R_{k,t}^{2^{p}}\right)k^{\ell-2^{p-2}}$ $\displaystyle\leq
C_{p}(\|\phi\|_{\gamma,\eta,H})\sum_{k=1}^{\infty}\mathbf{E}\sup_{k\leq t\leq
k+1}\exp(2^{p+1}\eta\|w_{0,t,h}\|^{2})k^{\ell-2^{p-2}}$ $\displaystyle\leq
C_{p}(\|\phi\|_{\gamma,\eta,H})\sum_{k=1}^{\infty}e^{2^{p+1}\eta\|w\|^{2}}k^{\ell-2^{p-2}}=C_{p}(\|\phi\|_{\gamma,\eta,H},\ell)e^{2^{p+1}\eta\|w\|^{2}},$

provided that $\ell<2^{p-2}-1$. The conclusion of Theorem 4.3 then follows
from the above estimates, the martingale approximation (4.14) and Lemma 4.10.
∎

####  4.3.2. The rate of convergence in CLT

We recall a Berry-Esseen type estimate for martingales from [22], which will
be used in the proof of the rate of convergence in the central limit theorem.

######  Theorem 4.18 (Theorem 3.10 of [22]).

Let $\displaystyle M_{N}=\sum_{j=1}^{N}Z_{j}$ be a zero mean martingale and
$\displaystyle\sigma_{k}^{2}=\sum_{j=1}^{k}\mathbf{E}Z_{j}^{2}$. If
$q>\frac{1}{2}$, and

(4.28) $\displaystyle\max_{j\leq
N}\frac{1}{\sigma_{N}^{4q}}\mathbf{E}|Z_{j}|^{4q}\leq\frac{M}{N^{2q}},$

for a constant $M>0$. Then there exists a constant $C$ depending only on $M$
and $q$ such that whenever

(4.29) $\displaystyle
N^{-q}+\mathbf{E}\left|\frac{1}{\sigma_{N}^{2}}\sum_{j=1}^{N}\mathbf{E}\left[Z_{j}^{2}|\mathcal{F}_{j-1}\right]-1\right|^{2q}\leq
1,$

one has

$\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left(\frac{M_{N}}{\sigma_{N}}\leq
z\right)-\Phi(z)\right|\leq
C\left(N^{-q}+\mathbf{E}\left|\frac{1}{\sigma_{N}^{2}}\sum_{j=1}^{N}\mathbf{E}\left[Z_{j}^{2}|\mathcal{F}_{j-1}\right]-1\right|^{2q}\right)^{1/(4q+1)},$

where $\Phi(z)$ is the distribution function of the standard normal
distribution.

We are now in a position to give an estimate of the convergence rate in the
central limit theorem.

###### Proof of Theorem 4.4.

Recall from (4.20) that the asymptotic variance

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

Since $Y\in C^{\overline{\gamma}_{0}}_{2^{5}\eta,H}(H\times\mathbb{T}^{n})$ by
Proposition 4.13, thus Proposition 4.16 implies that for
$\eta\in(0,2^{-7}p^{-1}\eta_{0}]$, and $N\geq 1$,

(4.30)
$\displaystyle\mathbf{E}_{(w,h)}\left|\frac{1}{N}\sum_{k=1}^{N}\Big{(}Y(X_{k-1})-\big{\langle}\Gamma_{\beta_{k-1}h},Y(\cdot,\beta_{k-1}h)\big{\rangle}\Big{)}\right|^{2p}\leq
C_{p}e^{2^{7}p\eta\|w\|^{2}}\|Y\|_{\overline{\gamma}_{0},2^{5}\eta,H}^{p}N^{-p}.$

In addition, since $F=\langle\Gamma_{h},Y(\cdot,h)\rangle\in
C^{\overline{\gamma}_{0}}(\mathbb{T}^{n},\mathbb{R})$ by Proposition 4.13, it
follows from Theorem 3 in [26] that for $N\geq 1$,

(4.31)
$\displaystyle\left|\frac{1}{N}\sum_{k=1}^{N}\big{\langle}\Gamma_{\beta_{k-1}h},Y(\cdot,\beta_{k-1}h)\big{\rangle}-\sigma^{2}\right|\leq
C\|F\|_{\overline{\gamma}_{0}}N^{-\frac{\overline{\gamma}_{0}}{A+n}},$

where $A$ is the constant from the Diophantine condition (2.2) and $n$ is the
dimension of the torus. Therefore, by the Markov property, inequality (4.30)
with $p=1$, and inequality (4.31), it follows that for
$\eta\in(0,2^{-7}\eta_{0}]$, and $N\geq 1$,

$\displaystyle\left|\frac{1}{N}\sigma_{N}^{2}-\sigma^{2}\right|=\left|\frac{1}{N}\sum_{k=1}^{N}\mathbf{E}_{(w,h)}Z_{k}^{2}-\sigma^{2}\right|$
$\displaystyle\begin{aligned}
\leq\Bigg{|}\frac{1}{N}\sum_{k=1}^{N}\Big{(}\mathbf{E}_{(w,h)}\left[\mathbf{E}_{(w,h)}\left[Z_{k}^{2}|\mathcal{F}_{k-1}\right]\right]&-\big{\langle}\Gamma_{\beta_{k-1}h},Y(\cdot,\beta_{k-1}h)\big{\rangle}\Big{)}\Bigg{|}\\\
&+\left|\frac{1}{N}\sum_{k=1}^{N}\big{\langle}\Gamma_{\beta_{k-1}h},Y(\cdot,\beta_{k-1}h)\big{\rangle}-\sigma^{2}\right|\end{aligned}$
$\displaystyle\leq\mathbf{E}_{(w,h)}\left|\frac{1}{N}\sum_{k=1}^{N}\Big{(}Y(X_{k-1})-\big{\langle}\Gamma_{\beta_{k-1}h},Y(\cdot,\beta_{k-1}h)\big{\rangle}\Big{)}\right|+\left|\frac{1}{N}\sum_{k=1}^{N}\big{\langle}\Gamma_{\beta_{k-1}h},Y(\cdot,\beta_{k-1}h)\big{\rangle}-\sigma^{2}\right|$
(4.32) $\displaystyle\leq
Ce^{2^{6}\eta\|w\|^{2}}\|Y\|_{\overline{\gamma}_{0},2^{5}\eta,H}^{1/2}N^{-1/2}+C\|F\|_{\overline{\gamma}_{0}}N^{-\frac{\overline{\gamma}_{0}}{A+n}}.$

This shows that $\frac{1}{N}\sigma_{N}^{2}$ converges to $\sigma^{2}$. To
estimate the convergence rate for the central limit theorem of the
approximating martingale sequence, we first deal with the case when
$\sigma^{2}>0$. Set $q=2^{p-2}$ for integer $p\geq 2$ in Theorem 4.18 and
choose $N_{1}>0$ such that for all $N\geq N_{1}$, inequality (4.29) is
satisfied and $\frac{1}{N}\sigma_{N}^{2}\in[\sigma^{2}/2,3\sigma^{2}/2]$. It
then follows from Lemma (4.9) that the condition (4.28) holds for $p\geq 2$
and $\eta\in(0,2^{-p-1}\eta_{0}]$.

Observe that

$\displaystyle\mathbf{E}_{(w,h)}\left|\frac{1}{\sigma_{N}^{2}}\sum_{j=1}^{N}\mathbf{E}_{(w,h)}\left[Z_{j}^{2}|\mathcal{F}_{j-1}\right]-1\right|^{2^{p-1}}$
$\displaystyle\leq
C_{p}\left(\mathbf{E}_{(w,h)}\left|\frac{1}{\sigma_{N}^{2}}\sum_{k=1}^{N}\Big{(}Y(X_{k-1})-F(\beta_{k-1}h)\Big{)}\right|^{2^{p-1}}+\left|\frac{1}{\sigma_{N}^{2}}\sum_{k=1}^{N}F(\beta_{k-1}h)-1\right|^{2^{p-1}}\right)$
$\displaystyle\leq
C_{p}\left(e^{2^{p+5}\eta\|w\|^{2}}\|Y\|_{\overline{\gamma}_{0},2^{5}\eta,H}^{2^{p-2}}N^{-2^{p-2}}+\left|\frac{1}{N}\sum_{k=1}^{N}F(\beta_{k-1}h)-\sigma^{2}\right|^{2^{p-1}}+\left|\frac{1}{N}\sigma_{N}^{2}-\sigma^{2}\right|^{2^{p-1}}\right)$
$\displaystyle\leq
C_{p}\left(e^{2^{p+5}\eta\|w\|^{2}}\|Y\|_{\overline{\gamma}_{0},2^{5}\eta,H}^{2^{p-2}}N^{-2^{p-2}}+\|F\|_{\overline{\gamma}_{0}}^{2^{p-1}}N^{-\frac{2^{p-1}\overline{\gamma}_{0}}{A+n}}\right).$

Therefore by Theorem 4.18 we have

$\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left(\frac{M_{N}}{\sigma_{N}}\leq
z\right)-\Phi(z)\right|$ $\displaystyle\leq
C\left(N^{-2^{p-2}}+\mathbf{E}\left|\frac{1}{\sigma_{N}^{2}}\sum_{j=1}^{N}\mathbf{E}\left[Z_{j}^{2}|\mathcal{F}_{j-1}\right]-1\right|^{2^{p-1}}\right)^{1/(2^{p}+1)}$
$\displaystyle\leq
Ce^{2^{5}\eta\|w\|^{2}}\left(N^{-\frac{2^{p-2}}{2^{p}+1}}+N^{-\frac{2^{p-1}\overline{\gamma}_{0}}{(2^{p}+1)(A+n)}}\right).$

As a result,

$\displaystyle\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left(\frac{M_{N}}{\sqrt{N}}\leq
z\right)-\Phi_{\sigma}(z)\right|$
$\displaystyle\leq\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left(\frac{M_{N}}{\sqrt{N}}\leq
z\right)-\Phi\left(\frac{\sqrt{N}}{\sigma_{N}}z\right)\right|+\sup_{z\in\mathbb{R}}\left|\Phi\left(\frac{\sqrt{N}}{\sigma_{N}}z\right)-\Phi_{\sigma}(z)\right|$
$\displaystyle\leq\sup_{z\in\mathbb{R}}\left|\mathbf{P}\left(\frac{M_{N}}{\sigma_{N}}\leq
z\right)-\Phi\left(z\right)\right|+C\left|\frac{\sigma_{N}}{\sqrt{N}}-\sigma\right|$
$\displaystyle\leq
Ce^{2^{5}\eta\|w\|^{2}}\left(N^{-\frac{2^{p-2}}{2^{p}+1}}+N^{-\frac{2^{p-1}\overline{\gamma}_{0}}{(2^{p}+1)(A+n)}}\right).$

When $\sigma=0$, we note that

$\displaystyle(|z|\wedge 1)\left|\mathbf{P}\left(\frac{M_{N}}{\sqrt{N}}\leq
z\right)-\Phi_{0}(z)\right|\leq(|z|\wedge
1)\left|\mathbf{P}\left(\left|\frac{M_{N}}{\sqrt{N}}\right|\geq|z|\right)\right|$
$\displaystyle\leq(|z|\wedge 1)|z|^{-1}N^{-1/2}\mathbf{E}_{(w,h)}|M_{N}|\leq
N^{-1/2}\left(\mathbf{E}_{(w,h)}|M_{N}|^{2}\right)^{1/2}=\left(\frac{1}{N}\sum_{k=1}^{N}Z_{k}^{2}\right)^{1/2}$
$\displaystyle\leq
Ce^{2^{5}\eta\|w\|^{2}}\left(N^{-1/4}+N^{-\frac{\overline{\gamma}_{0}}{2(A+n)}}\right)$

by estimate (4.32).

To pass the estimates to continuous times, we apply the following lemma from
[34].

######  Lemma 4.19.

Let $R_{1},R_{2}$ be real random variables. Then for any $\sigma\geq 0$ and
$\varepsilon>0$ we have

$\sup_{z\in\mathbb{R}}|\Delta_{\sigma}(R_{1},z)|\leq\sup_{z\in\mathbb{R}}|\Delta_{\sigma}(R_{2},z)|+\mathbf{P}(|R_{1}-R_{2}|>\varepsilon)+c_{\sigma}\varepsilon,$

where $c_{\sigma}$ is a constant depending only on $\sigma$ and
$\Delta_{\sigma}(R,z)$ for random variable $R$ is defined as

$\displaystyle\Delta_{\sigma}(R,z):=\left\\{\begin{array}[]{lr}\mathbf{P}(R\leq
z)-\Phi_{\sigma}(z)\text{ , }&\sigma>0,\\\ (|z|\wedge 1)\left(\mathbf{P}(R\leq
z)-\Phi_{0}(z)\right)\text{ , }&\sigma=0.\\\ \end{array}\right.$

Recall that $N$ is the integer part of $T$ in the martingale approximation
(4.14). It follows from the approximation that

$\displaystyle\left|\frac{1}{\sqrt{T}}\int_{0}^{T}\widetilde{\phi}(X_{t})dt-\frac{M_{N}}{\sqrt{N}}\right|\leq\frac{1}{\sqrt{NT}}\left|\int_{0}^{T}\widetilde{\phi}(X_{t})dt\right|+\frac{R_{N,T}}{\sqrt{N}}.$

The expectation of the remainder term $R_{N,T}$ obtained in the proof of Lemma
4.10 together with the Markov inequality yields

$\displaystyle\mathbf{P}\left(\frac{R_{N,T}}{\sqrt{N}}>\varepsilon/2\right)\leq
Ce^{2\eta\|w\|^{2}}N^{-4}\varepsilon^{-8}.$

It follows from (4.27) with $p=1$ that

$\displaystyle\mathbf{E}\left|\int_{0}^{T}\widetilde{\phi}(X_{t})dt\right|\leq\left(\mathbf{E}\left(\int_{0}^{T}\widetilde{\phi}(X_{t})dt\right)^{2}\right)^{\frac{1}{2}}\leq
T^{\frac{1}{2}}Ce^{2\eta\|w\|^{2}}.$

Now applying Lemma 4.19 with
$R_{1}=\frac{1}{\sqrt{T}}\int_{0}^{T}\widetilde{\phi}(X_{t})dt$ and
$R_{2}=\frac{M_{N}}{\sqrt{N}}$, and the Markov inequality we have

$\displaystyle\sup_{z\in\mathbb{R}}|\Delta_{\sigma}(R_{1},z)|$
$\displaystyle\leq\sup_{z\in\mathbb{R}}|\Delta_{\sigma}(R_{2},z)|+\mathbf{P}(|R_{1}-R_{2}|>\varepsilon)+c_{\sigma}\varepsilon$
$\displaystyle\leq\sup_{z\in\mathbb{R}}|\Delta_{\sigma}(R_{2},z)|+2\varepsilon^{-1}N^{-\frac{1}{2}}\mathbf{E}\left|T^{-\frac{1}{2}}\int_{0}^{T}\widetilde{\phi}(X_{t})dt\right|+Ce^{2\eta\|w\|^{2}}N^{-4}\varepsilon^{-8}+c_{\sigma}\varepsilon$
$\displaystyle\leq\sup_{z\in\mathbb{R}}|\Delta_{\sigma}(R_{2},z)|+Ce^{2\eta\|w\|^{2}}N^{-\frac{1}{4}}$

by taking $\varepsilon=N^{-\frac{1}{4}}$. ∎

##  5\. Large viscosity implies a trivial quasi-periodic attractor

In this section, we will show the existence of a unique stable quasi-periodic
solution of (2.3) when the viscosity is large as in Theorem 2.10. Recall that
$G=\sqrt{\|f\|_{\infty}^{2}/\nu^{4}+\mathcal{B}_{0}/\nu^{3}}$, and $c_{0}$ is
the constant from (2.20). Let $\delta_{0}=\nu-
c_{0}^{2}\nu^{-2}\left(\|f\|_{\infty}^{2}\nu^{-1}+\mathcal{B}_{0}\right)$.
Then $\delta_{0}>0$ is equivalent to $G<1/c_{0}$. Fix a lattice
$\mathbb{Z}_{\varkappa}=\\{\varkappa m:m\in\mathbb{Z}\\}$ where $\varkappa>0$
is a real number. For each $h\in\mathbb{T}^{n}$, denote by $\eqref{NS}_{h}$
the Navier-Stokes equation (2.3) with $f(t,x)=\Psi(\beta_{t}0,x)$ replaced by
$\Psi(\beta_{t}h,x)$. Let $n_{1}\in\mathbb{Z}_{\varkappa}$ and

$w_{N,h}(t,\omega)=w(n_{1}+N,t,h,\omega,0),\quad
N\in\mathbb{Z}_{\varkappa}^{-},t\geq n_{1}$

be the sequence of solutions of $\eqref{NS}_{h}$. In exactly the same way as
in [31, 30], one can show the following

######  Theorem 5.1.

Assume $\delta_{0}>0$ and let $\delta\in(0,\delta_{0})$. Then for each
$h\in\mathbb{T}^{n}$ there is a full measure subset $\Omega_{h,\varkappa}$ of
$\Omega$ such that:
1\. There is a complete random trajectory
$w^{*}(\cdot,h,\cdot):\mathbb{R}\times\Omega\rightarrow H$, which is a strong
solution of $\eqref{NS}_{h}$. For each $n_{1}\in\mathbb{Z}_{\varkappa}$ and
$\omega\in\Omega_{h,\varkappa}$, $w^{*}(\cdot,h,\omega)$ is the limit of
$\\{w_{N,h}(\cdot,\omega)\\}_{N\in\mathbb{Z}_{\varkappa}^{-}}$ in
$C([n_{1},\infty),H)$ equipped with the supremum norm.
2\. For any $s\in\mathbb{R}$, there exist positive random times
$n^{*}(s,\delta,h)$ and $n_{*}(s,\delta,h)$ having all moments finite such
that

$\displaystyle\sup_{{w_{0}}\in
B_{r}(w^{*}(s,h,\omega))}\|w(s,t,h,\omega,w_{0})-w^{*}(t,h,\omega)\|^{2}\leq
r^{2}e^{-\delta(t-s)},$ $\displaystyle\sup_{{w_{0}}\in
B_{r}(w^{*}(\tau,h,\omega))}\|w(\tau,s,h,\omega,w_{0})-w^{*}(s,h,\omega)\|^{2}\leq
r^{2}e^{-\delta(s-\tau)},$

for all $r>0$, $t>s+n^{*}$ and $\tau<s-n^{*}$. Here $B_{r}(w)$ is the ball
centered at $w$ with radius $r$ in $H$ with the norm $\|\cdot\|$.

The following proposition shows that there is a continuous modification of the
random field $w^{*}(0,h,\omega)$, which is proved by applying the Kolmogorov
continuity theorem. The desired quasi-periodic solution will be constructed
from this random field.

######  Proposition 5.2.

Assume $\Psi\in C^{\gamma}(\mathbb{T}^{n},H)$, $Gc_{0}\leq\sqrt{1/2}$. For any
$p\geq 1$, if

(5.1) $\displaystyle\nu^{3}>8pc_{0}^{2}\mathcal{B}_{0},$

then there is a constant $C>0$ such that

(5.2)
$\displaystyle\mathbf{E}\|w^{*}(0,h_{1},\cdot)-w^{*}(0,h_{2},\cdot)\|^{2p}\leq
C|h_{1}-h_{2}|^{p\gamma},\quad\forall h_{1},h_{2}\in\mathbb{T}^{n}.$

In particular, for any $\overline{\eta}>0$, if condition (5.1) holds for
$p=\frac{n+\overline{\eta}}{\gamma}$, then the random field
$w^{*}(0,h,\omega)$ has a continuous (with respect to $h$) modification, which
is $\eta$-Hölder continuous for all
$0<\eta<\frac{\overline{\eta}\gamma}{2(n+\overline{\eta})}$.

###### Proof.

For $N\in\mathbb{Z}_{\varkappa}^{-}$ and $h_{1},h_{2}\in\mathbb{T}^{n}$, let
$\mathcal{R}_{r}=w_{N,r,h_{1}}(0)-w_{N,r,h_{2}}(0)$. To show (5.2), we first
prove that the same inequality holds for $\mathcal{R}_{0}$ and then letting
$N\rightarrow-\infty$. Let $\delta>0$ be a constant whose value will be
determined later. Since $w_{N,r,h_{i}}(0)$ is the solution to
$\eqref{NS}_{h_{i}}$ starting from initial position $0\in H$ at initial time
$s=N$, it follows that (see also the proof of (6.3))

$\displaystyle\partial_{r}\|\mathcal{R}_{r}\|^{2}=\left\langle\nu\Delta\mathcal{R}_{r},2\mathcal{R}_{r}\right\rangle-\left\langle
B\left(\mathcal{K}\mathcal{R}_{r},w_{N,r,h_{1}}(0)\right),2\mathcal{R}_{r}\right\rangle+\left\langle\Psi(\beta_{r}h_{1})-\Psi(\beta_{r}h_{2}),2\mathcal{R}_{r}\right\rangle$
$\displaystyle\leq-2\nu\|\mathcal{R}_{r}\|_{1}^{2}+2c_{0}\|\mathcal{R}_{r}\|\|w_{N,r,h_{1}}(0)\|_{1}\|\mathcal{R}_{r}\|_{1}+2\|\Psi(\beta_{r}h_{1})-\Psi(\beta_{r}h_{2})\|\|\mathcal{R}_{r}\|$
$\displaystyle\leq-2\nu\|\mathcal{R}_{r}\|_{1}^{2}+c_{0}^{2}\nu^{-1}\|\mathcal{R}_{r}\|^{2}\|w_{N,r,h_{1}}(0)\|_{1}^{2}+\nu\|\mathcal{R}_{r}\|_{1}^{2}+4\delta^{-1}\|\Psi\|_{\gamma}|h_{1}-h_{2}|^{\gamma}+\frac{\delta}{4}\|\mathcal{R}_{r}\|^{2}$
$\displaystyle\leq-(\nu-\delta/4)\|\mathcal{R}_{r}\|^{2}+c_{0}^{2}\nu^{-1}\|\mathcal{R}_{r}\|^{2}\|w_{N,r,h_{1}}(0)\|_{1}^{2}+4\delta^{-1}\|\Psi\|_{\gamma}|h_{1}-h_{2}|^{\gamma}.$

By the Gronwall’s inequality and Hölder’s inequality, and noting that
$\mathcal{R}_{N}=0$, we have for $p,q>0$ with $1/p+1/q=1$,

$\displaystyle\|\mathcal{R}_{0}\|^{2p}$ $\displaystyle\leq
4^{p}\delta^{-p}\|\Psi\|_{\gamma}^{p}|h_{1}-h_{2}|^{p\gamma}\left(\int_{N}^{0}\exp{\left(\int_{z}^{0}-\left(\nu-\frac{\delta}{4}\right)+c_{0}^{2}\nu^{-1}\|w_{N,r,h_{1}}(0)\|_{1}^{2}dr\right)}dz\right)^{p}$
$\displaystyle\leq
4^{p}\delta^{-p}\|\Psi\|_{\gamma}^{p}|h_{1}-h_{2}|^{p\gamma}\left(\int_{N}^{0}e^{q\int_{z}^{0}-\frac{\delta}{4}dr}dz\right)^{\frac{p}{q}}\int_{N}^{0}e^{p\int_{z}^{0}-\left(\nu-\frac{\delta}{2}\right)+c_{0}^{2}\nu^{-1}\|w_{N,r,h_{1}}(0)\|_{1}^{2}dr}dz$
(5.3) $\displaystyle\leq
4^{2p-1}q^{-\frac{p}{q}}\delta^{-2p+1}\|\Psi\|_{\gamma}^{p}|h_{1}-h_{2}|^{p\gamma}\int_{N}^{0}e^{p\int_{z}^{0}-\left(\nu-\frac{\delta}{2}\right)+c_{0}^{2}\nu^{-1}\|w_{N,r,h_{1}}(0)\|_{1}^{2}dr}dz.$

We would like to have a bound on the expectation of
$\exp\left(\int_{z}^{0}pc_{0}^{2}\nu^{-1}\|w_{N,r,h_{1}}(0)\|_{1}^{2}dr\right)$.
It follows from Ito’s formula that for $N\leq z\leq t\leq 0$,

$\displaystyle\left\|w_{N,t,h_{1}}\right\|^{2}$
$\displaystyle+\nu\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr-\mathcal{B}_{0}(t-z)$
$\displaystyle=\left\|w_{N,z,h_{1}}\right\|^{2}+2\int_{z}^{t}\left\langle
w_{N,r,h_{1}},GdW(r)\right\rangle-\nu\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr+2\int_{z}^{t}\left\langle
w_{N,r,h_{1}},\Psi(\beta_{r}h_{1})\right\rangle dr$
$\displaystyle\leq\left\|w_{N,z,h_{1}}\right\|^{2}+2\int_{z}^{t}\left\langle
w_{N,r,h_{1}},GdW(r)\right\rangle+\frac{\|f\|_{\infty}^{2}}{\nu}(t-z).$

Therefore for $\varepsilon>0$,

$\displaystyle\frac{(1+\varepsilon)pc_{0}^{2}}{\nu}\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr-\frac{(1+\varepsilon)pc_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)(t$
$\displaystyle-z)\leq M(z,t),$

where
$M(z,t):=\frac{(1+\varepsilon)pc_{0}^{2}}{\nu^{2}}\left\|w_{N,z,h_{1}}\right\|^{2}+\frac{2(1+\varepsilon)pc_{0}^{2}}{\nu^{2}}\int_{z}^{t}\left\langle
w_{N,r,h_{1}},GdW(r)\right\rangle$ is a continuous square integrable
martingale whose quadratic variation $[M](z,t)$ satisfies

$\displaystyle[M](z,t)=\frac{4(1+\varepsilon)^{2}p^{2}c_{0}^{4}}{\nu^{4}}\int_{z}^{t}\sum_{k=1}^{d}\langle
w_{N,r,h_{1}},g_{k}\rangle^{2}dr\leq\frac{4(1+\varepsilon)^{2}p^{2}c_{0}^{4}\mathcal{B}_{0}}{\nu^{4}}\int_{z}^{t}\|w_{N,r,h_{1}}\|_{1}^{2}dr.$

As a consequence, one has

$\displaystyle\frac{pc_{0}^{2}}{\nu}\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr-\frac{(1+\varepsilon)pc_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)(t-z)$
$\displaystyle\leq M(z,t)-\frac{\varepsilon
pc_{0}^{2}}{\nu}\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr$
$\displaystyle\leq
M(z,t)-\frac{\varepsilon\nu^{3}}{4(1+\varepsilon)^{2}pc_{0}^{2}\mathcal{B}_{0}}[M](z,t).$

Let
$b=\frac{\varepsilon\nu^{3}}{2(1+\varepsilon)^{2}pc_{0}^{2}\mathcal{B}_{0}}$.
It then follows from the exponential supermartingale inequality that for
$K\geq 0$,

$\displaystyle\mathbf{P}\left(\sup_{t\geq
z}\exp\left(\frac{pc_{0}^{2}}{\nu}\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr-\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)(t-z)\right)\geq
e^{K}\middle|\mathcal{F}_{z}\right)$
$\displaystyle=\mathbf{P}\left(\sup_{t\geq
z}\left(\frac{pc_{0}^{2}}{\nu}\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr-\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)(t-z)\right)\geq
K\middle|\mathcal{F}_{z}\right)$ $\displaystyle\leq\mathbf{P}\left(\sup_{t\geq
z}\left(M(z,t)-\frac{b}{2}[M](z,t)\right)\geq K\middle|\mathcal{F}_{z}\right)$
$\displaystyle=\mathbf{P}\left(\sup_{t\geq
z}\exp\left(bM(z,t)-\frac{b^{2}}{2}[M](z,t)\right)\geq
e^{bK}\middle|\mathcal{F}_{z}\right)\leq\exp\left(\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left\|w_{N,z,h_{1}}\right\|^{2}\right)e^{-bK}.$

If
$b=\frac{\varepsilon\nu^{3}}{2(1+\varepsilon)^{2}pc_{0}^{2}\mathcal{B}_{0}}>1$,
then the same argument as we derive (6.7) implies that

(5.4) $\displaystyle\mathbf{E}\sup_{t\geq
z}\exp\left(\frac{pc_{0}^{2}}{\nu}\int_{z}^{t}\left\|w_{N,r,h_{1}}\right\|_{1}^{2}dr-\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)(t-z)\right)$
$\displaystyle\leq\mathbf{E}\exp\left(\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left\|w_{N,z,h_{1}}\right\|^{2}\right)\frac{4}{1-2^{1-b}}.$

To bound the expectation on the right, we note that the solution
$w_{N,r,h_{1}}$ starts from $0$, so (6.6) implies that

$\mathbf{P}\left(\left\|w_{N,z,h_{1}}\right\|^{2}-\frac{C(f,\mathcal{B}_{0})}{\nu}>\frac{K}{\alpha}\right)\leq
e^{-K},$

where $\alpha=\frac{(1-a)\nu}{2\mathcal{B}_{0}}$,
$C(f,\mathcal{B}_{0})=\frac{\|f\|_{\infty}^{2}}{2a}+\mathcal{B}_{0}$ and
$a\in(0,1)$. Therefore

$\mathbf{P}\left(\exp\left(\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left\|w_{N,z,h_{1}}\right\|^{2}-\frac{(\varepsilon+1)pc_{0}^{2}C(f,\mathcal{B}_{0})}{\nu^{3}}\right)>e^{\frac{(\varepsilon+1)pc_{0}^{2}K}{\alpha\nu^{2}}}\right)\leq
e^{-K},$

Again by the same argument as in (6.7) with
$c=\frac{\alpha\nu^{2}}{(\varepsilon+1)pc_{0}^{2}}$, one has

(5.5)
$\displaystyle\mathbf{E}\exp\left(\frac{(\varepsilon+1)pc_{0}^{2}}{\nu^{2}}\left\|w_{N,z,h_{1}}\right\|^{2}-\frac{(\varepsilon+1)pc_{0}^{2}C(f,\mathcal{B}_{0})}{\nu^{3}}\right)\leq\frac{4}{1-2^{1-c}},$

provided that
$c=\frac{\alpha\nu^{2}}{(\varepsilon+1)pc_{0}^{2}}=\frac{(1-a)\nu^{3}}{2(\varepsilon+1)pc_{0}^{2}\mathcal{B}_{0}}>1$.
We take $a=\frac{1}{1+\varepsilon}$ , which yields $b=c$.

Now it follows from (5.3)-(5.5) that

$\displaystyle\mathbf{E}\|\mathcal{R}_{0}\|^{2p}$ $\displaystyle\leq
4^{2p-1}q^{-\frac{p}{q}}\delta^{-2p+1}\|\Psi\|_{\gamma}^{p}|h_{1}-h_{2}|^{p\gamma}\int_{N}^{0}e^{p\left(\nu-\frac{\delta}{2}\right)z}\mathbf{E}e^{p\int_{z}^{0}c_{0}^{2}\nu^{-1}\|w_{N,r,h_{1}}(0)\|_{1}^{2}dr}dz$
$\displaystyle\leq
C(\delta,\varepsilon)|h_{1}-h_{2}|^{p\gamma}\int_{N}^{0}e^{p\left(\nu-\frac{\delta}{2}\right)z}e^{-\frac{(1+\varepsilon)pc_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)z}dz$
(5.6)
$\displaystyle=C(\delta,\varepsilon)|h_{1}-h_{2}|^{p\gamma}\int_{N}^{0}e^{pz\left(\nu-\frac{\delta}{2}-\frac{(1+\varepsilon)c_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)\right)}dz$

where
$C(\delta,\varepsilon)=\left(\frac{1}{1-2^{1-c}}\right)^{2}4^{2p+1}q^{-\frac{p}{q}}\delta^{-2p+1}\|\Psi\|_{\gamma}^{p}\exp\left(\frac{(1+\varepsilon)pc_{0}^{2}C(f,\mathcal{B}_{0})}{\nu^{3}}\right).$
Since
$\delta_{0}=\nu-\frac{c_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)>0$,
one has

$\delta_{\varepsilon}:=\nu-\frac{(1+\varepsilon)c_{0}^{2}}{\nu^{2}}\left(\mathcal{B}_{0}+\frac{\|f\|_{\infty}^{2}}{\nu}\right)>0,\quad\text{as
long as }\quad\varepsilon<\varepsilon_{0}:=(Gc_{0})^{-2}-1.$

Keeping in mind that we need to ensure inequalities (5.4) and (5.5) hold,
which amounts to show the existence of $\varepsilon\in(0,\varepsilon_{0})$
such that
$c=\frac{\varepsilon\nu^{3}}{2(1+\varepsilon)^{2}pc_{0}^{2}\mathcal{B}_{0}}>1$.
This can be achieved if we let $(Gc_{0})^{2}\leq\frac{1}{2}$ (thus
$\varepsilon_{0}\geq 1$), and

$\frac{2pc_{0}^{2}\mathcal{B}_{0}}{\nu^{3}}<\sup_{\varepsilon\in(0,\varepsilon_{0})}\frac{\varepsilon}{(1+\varepsilon)^{2}}=\frac{1}{4}.$

Therefore, under the conditions given in Proposition 5.2, there is an
$\overline{\varepsilon}\in(0,\varepsilon_{0})$, such that
$\delta_{\overline{\varepsilon}}>0$. And by choosing
$\delta=\delta_{\overline{\varepsilon}}$, we have from (5.6) that

$\displaystyle\mathbf{E}\|\mathcal{R}_{0}\|^{2p}\leq
C|h_{1}-h_{2}|^{p\gamma},\quad\text{with}\quad
C=\frac{2C(\delta_{\overline{\varepsilon}},\overline{\varepsilon})}{p\delta_{\overline{\varepsilon}}}.$

By Theorem 5.1, we know that for $i=1,2$, as $N\rightarrow-\infty$,
$w_{N,r,h_{i}}(0)$ converges to $w^{*}(0,h_{i},\omega)$ almost surely. Hence
it follows from Fatou’s lemma that

(5.7)
$\displaystyle\mathbf{E}\|w^{*}(0,h_{1},\cdot)-w^{*}(0,h_{2},\cdot)\|^{2p}\leq\liminf_{N\rightarrow-\infty}\mathbf{E}\|\mathcal{R}_{0}\|^{2p}\leq
C|h_{1}-h_{2}|^{p\gamma}.$

Since the dimension of the torus is $n$, by taking
$p=\frac{n+\overline{\eta}}{\gamma}$ for any $\overline{\eta}>0$, we have

$\mathbf{E}\|w^{*}(0,h_{1},\cdot)-w^{*}(0,h_{2},\cdot)\|^{\frac{2(n+\overline{\eta})}{\gamma}}\leq
C|h_{1}-h_{2}|^{n+\overline{\eta}},\quad\forall h_{1},h_{2}\in\mathbb{T}^{n}.$

Hence by Kolmogorov’s continuity test, the random field $w^{*}(0,h,\omega)$
has a continuous modification. Moreover, with probability one, the
modification is $\eta$-Hölder continuous for all
$0<\eta<\frac{\overline{\eta}\gamma}{2(n+\overline{\eta})}$. ∎

Note that for $h=0\in\mathbb{T}^{n}$, the associated equation $\eqref{NS}_{0}$
is actually (2.3). We will show that the complete trajectory
$w^{*}(t,0,\omega)$ given in Theorem 5.1 is a quasi-periodic solution in the
sense of Definition 2.5.

######  Proposition 5.3.

The process $w^{*}(t,0,\omega)$ is a quasi-periodic solution of (2.3).

###### Proof.

The invariance property in Definition 2.5 follows from the stochastic flow
property and Theorem 5.1. Note that for any $\tau\geq 0$ and $s\in\mathbb{R}$,

$\displaystyle w^{*}(s+\tau,0,\omega)$
$\displaystyle=\lim_{\begin{subarray}{c}N\rightarrow\infty\\\
N\in\mathbb{Z}_{\varkappa}\end{subarray}}w(n_{1}-N,s+\tau,0,\omega,0)$
$\displaystyle=\lim_{\begin{subarray}{c}N\rightarrow\infty\\\
N\in\mathbb{Z}_{\varkappa}\end{subarray}}\Phi(s+\tau,\omega;s,w(n_{1}-N,s,0,\omega,0))$
$\displaystyle=\Phi(s+\tau,\omega;s,\lim_{\begin{subarray}{c}N\rightarrow\infty\\\
N\in\mathbb{Z}_{\varkappa}\end{subarray}}w(n_{1}-N,s,0,\omega,0))=\Phi(s+\tau,\omega;s,w^{*}(s,0,\omega)),\quad\mathbf{P}\text{-}\mathrm{a.s}..$

For $t\in\mathbb{Z}_{\varkappa}$, it follows that

$\displaystyle w^{*}(t,0,\omega)$
$\displaystyle=\lim_{\begin{subarray}{c}N\rightarrow\infty\\\
N\in\mathbb{Z}_{\varkappa}\end{subarray}}w(n_{1}-N,t,0,\omega,0)$
$\displaystyle=\lim_{\begin{subarray}{c}N\rightarrow\infty\\\
N\in\mathbb{Z}_{\varkappa}\end{subarray}}w(n_{1}-N-t,0,\beta_{t}0,\theta_{t}\omega,0)=w^{*}(0,\beta_{t}0,\theta_{t}\omega),\quad\mathbf{P}\text{-}\mathrm{a.s}..$

In particular, for each $\varkappa\in\mathbb{R}$, there is a subset
$\Omega_{\varkappa}\subset\Omega$ of full measure such that

$w^{*}(\varkappa,0,\omega)=w^{*}(0,\beta_{\varkappa}0,\theta_{\varkappa}\omega),\quad\forall\omega\in\Omega_{\varkappa}.$

Therefore for each $t\in\mathbb{R}$,

$w^{*}(t,0,\theta_{-t}\omega)=w^{*}(0,\beta_{t}0,\omega),\quad\mathbf{P}\text{-}\mathrm{a.s}..$

By Proposition 5.2, we can choose a continuous version of $w^{*}(0,h,\omega)$
such that $w^{*}(t,0,\theta_{-t}\omega)$ is a random quasi-periodic function
in the sense of Definition 2.5. This completes the proof. ∎

##  6\. Appendix: Estimates of the solution

Several estimates about the solution $w_{s,t,h}(w_{0})$ of the stochastic
Navier-Stokes equation (2.3) with time symbol $h\in\mathbb{T}^{n}$ are
collected in the following Lemma 6.1. Note that for any $h\in\mathbb{T}^{n}$,
we have

$\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h)\|=\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}0)\|=\sup_{t\in\mathbb{R}}\|f(t)\|:=\|f\|_{\infty},$

therefore the constant $C$ in bounds on the solution does not depend on $h$.

######  Lemma 6.1.

Let $a\in(0,1)$, $c\in(1,\infty)$,
$\eta_{0}=\frac{(1-a)\nu}{2c\mathcal{B}_{0}}$ and
$C(f,\mathcal{B}_{0})=\frac{\|f\|_{\infty}^{2}}{a\nu}+\mathcal{B}_{0}$. For
solutions of the Navier-Stokes equation (2.3), we have

  1. (1)

For every $t>s,h\in\mathbb{T}^{n}$ and every $\eta\in\left(0,\eta_{0}\right],$
we have

(6.1) $\mathbf{E}\exp\left(\eta\left\|w_{s,t,h}\right\|^{2}\right)\leq
C\exp\left(\eta e^{-\nu(t-s)}\left\|w_{0}\right\|^{2}\right),$

where
$C=\frac{4}{1-2^{1-c}}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)$.

  2. (2)

The following inequality

(6.2)
$\displaystyle\quad\mathbf{E}\exp\left(\eta\sup_{t\geq\tau}\left(\left\|w_{s,t,h}\right\|^{2}+\nu\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr-C(f,\mathcal{B}_{0})(t-\tau)\right)\right)\leq
C\exp\left(\eta e^{-\nu\left(\tau-s\right)}\left\|w_{0}\right\|^{2}\right)$

holds for every $\tau\geq s$ and $\eta\in\left(0,\eta_{0}\right]$, where
$C=\frac{16}{(1-2^{1-c})^{2}}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)$.

  3. (3)

For any $h_{1},h_{2}\in\mathbb{T}^{n}$, and every
$\eta\in\left(0,\eta_{0}\right]$, with $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta
C(f,\mathcal{B}_{0})$ and
$\overline{C}=\frac{16(r\nu)^{-1}}{(1-2^{1-c})^{2}}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)$,
we have

(6.3)
$\displaystyle\mathbf{E}\left\|w_{s,t,h_{1}}-w_{s,t,h_{2}}\right\|^{2}\leq
Ce^{r(t-s)}\exp\left({\eta\|w_{0}\|^{2}}\right)\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{1})-\Psi(\beta_{t}h_{2})\|^{2},$

for every $t\geq s$, where for $i=1,2$, $w_{s,t,h_{i}}$ is the solution to
equation (2.3) with $f=\Psi(\beta_{t}0)$ replaced by $\Psi(\beta_{t}h_{i})$
and with initial condition $(s,w_{0})$.

  4. (4)

For any $h\in\mathbb{T}^{n}$, and every $\eta\in\left(0,\eta_{0}\right]$, we
have

(6.4)
$\displaystyle\mathbf{E}\|w_{s,s+t,h}(w_{1})-w_{s,s+t,h}(w_{2})\|^{2}\leq
C\|w_{1}-w_{2}\|^{2}e^{\eta\|w_{1}\|^{2}+r(t-s)},$

for every $s\in\mathbb{R}$ and $t\geq 0$. Here
$C=64(1-2^{1-c})^{-3/2}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)$
and $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$.

###### Proof.

  1. (1)

Applying Ito’s formula to the functional $F(t,w)=e^{\nu(t-s)}\|w\|^{2}$ and
noting the fact that $\langle B(\mathcal{K}w,w),w\rangle=0$, we have

$\displaystyle e^{\nu(t-s)}\left\|w_{s,t,h}\right\|^{2}=$
$\displaystyle-\left\|w_{0}\right\|^{2}-2\nu\int_{s}^{t}e^{\nu(r-s)}\left\|w_{s,r,h}\right\|_{1}^{2}dr+2\int_{s}^{t}e^{\nu(r-s)}\left\langle
w_{s,r,h},\sum_{k=1}^{d}g_{k}dW_{k}(r)\right\rangle$
$\displaystyle+\nu\int_{s}^{t}e^{\nu(r-s)}\left\|w_{s,r,h}\right\|^{2}dr+2\int_{s}^{t}e^{\nu(r-s)}\left\langle
w_{s,r,h},\Psi(\beta_{r}h)\right\rangle
dr+\frac{e^{\nu(t-s)}-1}{\nu}\mathcal{B}_{0}.$

Given $0<a<1$, let
$C(f,\mathcal{B}_{0})=\frac{\|f\|_{\infty}^{2}}{a\nu}+\mathcal{B}_{0}$. By the
inequality $\|w\|_{1}^{2}\geq\|w\|^{2}$ and Young’s product inequality, it
follows that

$\displaystyle\left\|w_{s,t,h}\right\|^{2}$
$\displaystyle-e^{-\nu(t-s)}\|w_{0}\|^{2}-\frac{C(f,\mathcal{B}_{0})}{\nu}$
$\displaystyle\leq-(1-a)\nu\int_{s}^{t}e^{-\nu(t-r)}\|w_{s,r,h}\|^{2}dr+2\int_{s}^{t}e^{\nu(r-s)}\left\langle
w_{s,r,h},\sum_{k=1}^{d}g_{k}dW_{k}(r)\right\rangle.$

Let $M_{t}=2\int_{s}^{t}\left\langle
w_{s,r,h},\sum_{k=1}^{d}g_{k}dW_{k}(r)\right\rangle$ whose quadratic variation
is

$[M]_{t}=4\int_{s}^{t}\sum_{k=1}^{d}\left\langle
w_{s,r,h},g_{k}\right\rangle^{2}dr.$

Observe that

(6.5)
$\displaystyle-(1-a)\nu\|w\|^{2}\leq-\frac{(1-a)\nu}{\mathcal{B}_{0}}\sum_{k=1}^{d}\left\langle
w,g_{k}\right\rangle^{2}=-\frac{\alpha}{2}\sum_{k=1}^{d}4\left\langle
w,g_{k}\right\rangle^{2}$

where $\alpha=\frac{(1-a)\nu}{2\mathcal{B}_{0}}$. Hence

$\displaystyle\left\|w_{s,t,h}\right\|^{2}-e^{-\nu(t-s)}\|w_{0}\|^{2}-\frac{C(f,\mathcal{B}_{0})}{\nu}\leq\int_{s}^{t}e^{-\nu(t-r)}dM_{r}-\frac{\alpha}{2}\int_{s}^{t}e^{-\nu(t-r)}d[M]_{r}.$

Then Lemma A.1 from [32] implies that

(6.6)
$\displaystyle\mathbf{P}\left(\left\|w_{s,t,h}\right\|^{2}-e^{-\nu(t-s)}\left\|w_{0}\right\|^{2}-\frac{C(f,\mathcal{B}_{0})}{\nu}>\frac{K}{\alpha}\right)\leq
e^{-K},$

which is, for any $c>1$ and with $\eta_{0}=\frac{\alpha}{c}$, equivalent to

$\mathbf{P}\left(\exp\left(\eta_{0}\left\|w_{s,t,h}\right\|^{2}-\eta_{0}e^{-\nu(t-s)}\left\|w_{0}\right\|^{2}-\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)>e^{\frac{K}{c}}\right)\leq
e^{-K}.$

Now if a random variable $X$ satisfies $\mathbf{P}\left(X\geq
C\right)\leq\frac{1}{C^{c}}$ for every $C\geq 1$, then

$\displaystyle\mathbf{E}X=\int_{\mathrm{\Omega}}Xd\mathbf{P}$
$\displaystyle\leq\int_{\\{0\leq X\leq 1\\}}Xd\mathbf{P}+\int_{\\{X\geq
1\\}}Xd\mathbf{P}\leq 1+\sum_{n=0}^{\infty}\int_{\\{2^{n}\leq X\leq
2^{n+1}\\}}Xd\mathbf{P}$ (6.7) $\displaystyle\leq
1+\sum_{n=0}^{\infty}2^{n+1}\frac{1}{2^{cn}}\leq\frac{4}{1-2^{1-c}}.$

Therefore we have for $\eta\in(0,\eta_{0}]$, by Hölder’s inequality,

$\displaystyle\mathbf{E}\exp\left(\eta\left\|w_{s,t,h}\right\|^{2}-\eta
e^{-\nu(t-s)}\left\|w_{0}\right\|^{2}-\frac{\eta
C(f,\mathcal{B}_{0})}{\nu}\right)$
$\displaystyle\leq\left(\mathbf{E}\exp\left(\eta_{0}\left\|w_{s,t,h}\right\|^{2}-\eta_{0}e^{-\nu(t-s)}\left\|w_{0}\right\|^{2}-\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)\right)^{\eta/\eta_{0}}\leq\frac{4}{1-2^{1-c}}.$

Hence we arrive at (6.1) with
$C=\frac{4}{1-2^{1-c}}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)$
and $\eta_{0}=\frac{\alpha}{c}=\frac{(1-a)\nu}{2c\mathcal{B}_{0}}$.

  2. (2)

Again apply Ito’s formula, for any $\eta>0$, $s\leq\tau<t$,

$\displaystyle\eta\left\|w_{s,t,h}\right\|^{2}$
$\displaystyle+\eta\nu\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr-\eta\mathcal{B}_{0}(t-\tau)$
$\displaystyle=\eta\left\|w_{s,\tau,h}\right\|^{2}+2\eta\int_{\tau}^{t}\left\langle
w_{s,r,h},GdW(r)\right\rangle-\eta\nu\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr+2\eta\int_{\tau}^{t}\left\langle
w_{s,r,h},\Psi(\beta_{r}h)\right\rangle dr$ (6.8)
$\displaystyle\leq\eta\left\|w_{s,\tau,h}\right\|^{2}+2\eta\int_{\tau}^{t}\left\langle
w_{s,r,h},GdW(r)\right\rangle-\eta\nu(1-a)\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr+\frac{\eta}{a\nu}\|f\|_{\infty}^{2}\left(t-\tau\right)$

Therefore

$\displaystyle\eta\left\|w_{s,t,h}\right\|^{2}$
$\displaystyle+\eta\nu\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr-\eta
C(f,\mathcal{B}_{0})(t-\tau)$
$\displaystyle\leq\eta\left\|w_{s,\tau,h}\right\|^{2}+2\eta\int_{\tau}^{t}\left\langle
w_{s,r,h},GdW(r)\right\rangle-\eta\nu(1-a)\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr$
(6.9)
$\displaystyle\leq\eta\left\|w_{s,\tau,h}\right\|^{2}+2\eta\int_{\tau}^{t}\left\langle
w_{s,r,h},GdW(r)\right\rangle-\frac{\eta\alpha}{2}\int_{\tau}^{t}\sum_{k=1}^{d}4\left\langle
w_{s,r,h},g_{k}\right\rangle^{2}dr$

Again the last inequality is obtained by choosing
$\alpha=\frac{(1-a)\nu}{2\mathcal{B}_{0}}$ as in (6.5). Setting
$M(\tau,t)=\eta\left\|w_{s,\tau,h}\right\|^{2}+2\eta\int_{\tau}^{t}\left\langle
w_{s,r,h},GdW(r)\right\rangle$, then the right hand side of inequality (6.9)
is $M(\tau,t)-\frac{\alpha}{2\eta}[M](\tau,t)$, where $[M](\tau,t)$ is the
quadratic variation of the continuous $\mathrm{L}^{2}$-martingale $M$. Hence
by the exponential supermartingale inequality, it follows that

$\mathbf{P}\left(\sup_{t\geq\tau}\left(M(\tau,t)-\frac{\alpha}{2\eta}[M](\tau,t)\right)\geq
K\,\middle|\,\mathcal{F}_{\tau}\right)\leq\exp\left(\eta\left\|w_{s,\tau,h}\right\|^{2}-\frac{\alpha
K}{\eta}\right)$

for all $\tau\geq s$. As a consequence,

(6.10)
$\displaystyle\mathbf{P}\Big{(}\eta\sup_{t\geq\tau}\Big{(}\left\|w_{s,t,h}\right\|^{2}+\nu\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr$
$\displaystyle-C(f,\mathcal{B}_{0})(t-\tau)\Big{)}\geq
K\,\Big{|}\,\mathcal{F}_{\tau}\Big{)}$
$\displaystyle\leq\exp\Big{(}\eta\left\|w_{s,\tau,h}\right\|^{2}-\frac{\alpha
K}{\eta}\Big{)}.$

In view of (6.9) and (6.7), we deduce that

$\mathbf{E}\exp\left(\frac{\alpha}{c}\sup_{t\geq\tau}\left(\left\|w_{s,t,h}\right\|^{2}+\nu\int_{\tau}^{t}\left\|w_{s,r,h}\right\|_{1}^{2}dr-C(f,\mathcal{B}_{0})(t-\tau)\right)\right)\leq\frac{4}{1-2^{1-c}}\mathbf{E}\exp\left(\eta\|w_{s,\tau,h}\|^{2}\right)$

The conclusion follows from (6.1) by taking $\eta_{0}=\frac{\alpha}{c}$.

  3. (3)

Let $\mathcal{R}_{t}=w_{s,t,h_{1}}-w_{s,t,h_{2}}$ then

$\displaystyle\partial_{t}\mathcal{R}_{t}=$
$\displaystyle\nu\Delta\mathcal{R}_{t}+B\left(\mathcal{K}w_{s,t,h_{2}},w_{s,t,h_{2}}\right)-B\left(\mathcal{K}w_{s,t,h_{1}},w_{s,t,h_{1}}\right)+\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})$
$\displaystyle=\nu\Delta\mathcal{R}_{t}+B\left(\mathcal{K}\mathcal{R}_{t},\mathcal{R}_{t}\right)-B\left(\mathcal{K}\mathcal{R}_{t},w_{s,t,h_{1}}\right)-B\left(\mathcal{K}w_{s,t,h_{1}},\mathcal{R}_{t}\right)+\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1}).$

Therefore from the inequality $\left|\langle
B(\mathcal{K}u,v),w\rangle\right|\leq c_{0}\|u\|\|v\|_{1}\|w\|_{1/2}$ and the
interpolation inequality
$\|w\|_{1/2}^{2}\leq\varepsilon\|w\|_{1}^{2}+\varepsilon^{-2}\|w\|^{2}$ for
$\varepsilon>0$, we have

$\displaystyle\partial_{t}\|\mathcal{R}_{t}\|^{2}=\left\langle\nu\Delta\mathcal{R}_{t},2\mathcal{R}_{t}\right\rangle-\left\langle
B\left(\mathcal{K}\mathcal{R}_{t},w_{s,t,h_{1}}\right),2\mathcal{R}_{t}\right\rangle+\left\langle\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1}),2\mathcal{R}_{t}\right\rangle$
$\displaystyle\leq-2\nu\|\mathcal{R}_{t}\|_{1}^{2}+2c_{0}\|\mathcal{R}_{t}\|\|w_{s,t,h_{1}}\|_{1}\|\mathcal{R}_{t}\|_{1/2}+2\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})\|\|\mathcal{R}_{t}\|$
$\displaystyle\leq-2\nu\|\mathcal{R}_{t}\|_{1}^{2}+\frac{4c_{0}^{2}}{\eta\nu}\|\mathcal{R}_{t}\|_{1}^{2}+\eta\nu\|w_{s,t,h_{1}}\|_{1}^{2}\|\mathcal{R}_{t}\|^{2}+\nu\|\mathcal{R}_{t}\|_{1}^{2}+\frac{1}{\nu}\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})\|^{2}$
$\displaystyle\leq\left(C(\eta,\nu)+\eta\nu\|w_{s,t,h_{1}}\|_{1}^{2}\right)\|\mathcal{R}_{t}\|^{2}+\frac{1}{\nu}\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})\|^{2},$

where $C(\eta,\nu)=64c_{0}^{6}\eta^{-3}\nu^{-5}$. By Gronwall’s inequality,
and estimate (6.2), we have

$\displaystyle\mathbf{E}\|\mathcal{R}_{t}\|^{2}$
$\displaystyle\leq\frac{1}{\nu}\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})\|^{2}\int_{s}^{t}\exp\Big{(}C(\eta,\nu)(t-\tau)\Big{)}\mathbf{E}\left[\exp\left(\int_{\tau}^{t}\eta\nu\|w_{s,r,h_{1}}\|_{1}^{2}\right)dr\right]d\tau$
$\displaystyle\leq\frac{C}{\nu}\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})\|^{2}\int_{s}^{t}\exp\left(r(t-\tau)\right)\exp\left(\eta
e^{-\nu(\tau-s)}\|w_{0}\|^{2}\right)d\tau$
$\displaystyle\leq\overline{C}e^{r(t-s)}\exp\left(\eta\|w_{0}\|^{2}\right)\sup_{t\in\mathbb{R}}\|\Psi(\beta_{t}h_{2})-\Psi(\beta_{t}h_{1})\|^{2},$

with $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$ and
$\overline{C}=\frac{16(r\nu)^{-1}}{(1-2^{1-c})^{2}}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right).$

  4. (4)

We now compare solutions that start from different initial position. Let
$w_{0},\widetilde{w}_{0}\in H$, and
$\mathfrak{e}_{t}=w_{s,t,h}(w_{0})-\widetilde{w}_{s,t,h}(\widetilde{w}_{0})$,
where $w_{s,t,h}(w_{0}),\widetilde{w}_{s,t,h}(\widetilde{w}_{0})$ are the
solutions starting from $w_{0},\widetilde{w}_{0}$. In view of equation (2.3)
with symbol $h$, we see that $\mathfrak{e}_{t}$ solves the following equation

$\displaystyle\partial_{t}\mathfrak{e}_{t}$
$\displaystyle=\nu\mathrm{\Delta}\mathfrak{e}_{t}-B(\mathcal{K}w_{s,t,h},w_{s,t,h})+B(\mathcal{K}\widetilde{w}_{s,t,h},\widetilde{w}_{s,t,h})$
$\displaystyle=\nu\mathrm{\Delta}\mathfrak{e}_{t}-B(\mathcal{K}w_{s,t,h},w_{s,t,h})+B(\mathcal{K}(w_{s,t,h}-\mathfrak{e}_{t}),w_{s,t,h}-\mathfrak{e}_{t})$
$\displaystyle=\nu\mathrm{\Delta}\mathfrak{e}_{t}-B(\mathcal{K}w_{s,t,h},\mathfrak{e}_{t})+B(\mathcal{K}\mathfrak{e}_{t},w_{s,t,h})+B(\mathcal{K}\mathfrak{e}_{t},\mathfrak{e}_{t}).$

From the fact $\langle B(\mathcal{K}w,v),v\rangle=0$ and the basic estimates
of the nonlinear term as in the proof of (6.3), we have

$\displaystyle\partial_{t}\|\mathfrak{e}_{t}\|^{2}$
$\displaystyle=2\langle\mathfrak{e}_{t},\partial_{t}\mathfrak{e}_{t}\rangle=-2\nu\|\mathfrak{e}_{t}\|_{1}^{2}+2\langle
B(\mathcal{K}\mathfrak{e}_{t},w_{s,t,h}),\mathfrak{e}_{t}\rangle$
$\displaystyle\leq-2\nu\|\mathfrak{e}_{t}\|_{1}^{2}+2c_{0}\|\mathfrak{e}_{t}\|\|w_{s,t,h}\|_{1}\|\mathfrak{e}_{t}\|_{1/2}$
$\displaystyle\leq\left(C(\eta,\nu)+\eta\nu\|w_{s,t,h_{1}}\|_{1}^{2}\right)\|\mathfrak{e}_{t}\|^{2},$

where $C(\eta,\nu)=64c_{0}^{6}\eta^{-3}\nu^{-5}$. Hence by Gronwall’s
inequality,

$\displaystyle\|\mathfrak{e}_{t}\|^{2}\leq\|\mathfrak{e}_{s}\|^{2}\exp\left(C(\eta,\nu)(t-s)+\eta\nu\int_{s}^{t}\|w_{s,r,h_{1}}\|_{1}^{2}dt\right).$

From the estimate (6.2), we have

(6.11)
$\displaystyle\mathbf{E}\|w_{s,t,h}(w_{0})-\widetilde{w}_{s,t,h}(\widetilde{w}_{0})\|^{2}\leq
C\|w_{0}-\widetilde{w}_{0}\|^{2}e^{\eta\|w_{0}\|^{2}+r(t-s)},$

where
$C=\frac{16}{(1-2^{1-c})^{2}}\exp\left(\frac{\eta_{0}C(f,\mathcal{B}_{0})}{\nu}\right)$
and $r=64c_{0}^{6}\eta^{-3}\nu^{-5}+\eta C(f,\mathcal{B}_{0})$.

∎

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