are known as the strong dissipative conditions that imply the existence of a unique invariant measure and moreover, it has exponentially mixing property for the Markov semigroup associated to the fast variable. ]{}

By Theorem \[mixuniq\], it is easy to prove that Eq. (\[spde-fs\]) has a unique mild pathwise solution. Here, we omit the proof.

Suppose that conditions [(A1)-(A3)]{} are satisfied. Then, for any initial values $X_{0},Y_0\in V_\beta,\beta>\alpha$, Eq. (\[spde-fs\]) has a unique mild pathwise solution $(X^\varepsilon_t,Y^\varepsilon_t)$, i.e., $$\begin{aligned}
\label{mild2}
X^{\varepsilon}_t
&=S_t X^{\varepsilon}_{0}+\int_0^tS_{t-s}b(X^{\varepsilon}_s,Y^{\varepsilon}_s)\mathrm{d}s+\int_0^tS_{t-s}g(X^{\varepsilon}_s)\mathrm{d}B^H_s, \qquad X^{\varepsilon}_0= X_0, \\
\label{mild3} Y^{\varepsilon}_t& =S_\frac{t}{\varepsilon}Y^{\varepsilon}_{0}+\frac{1}{\varepsilon}\int_0^tS_{\frac{t-s}{\varepsilon}}F(X^{\varepsilon}_s,Y^{\varepsilon}_s)\mathrm{d}s+\frac{1}{\sqrt{\varepsilon}}\int_0^tS_{\frac{t-s}{\varepsilon}}G(X^{\varepsilon}_s,Y^{\varepsilon}_s)\mathrm{d}W_s, \qquad  Y^{\varepsilon}_0= Y_0.\end{aligned}$$

Denote by $\bar{X}_t, t\in[0,T]$, the solution of the following SPDEs driven by fBm, $$\begin{aligned}
\label{ave-eq}
\begin{split}
\mathrm{d} \bar{X}_t=(A\bar{X}_t+\bar{b}(\bar{X}_t))\mathrm{d}t+g(\bar{X}_t)\mathrm{d}B^H_t,  \qquad \bar{X}_0= X_0,
\end{split}\end{aligned}$$ where $$\begin{aligned}
\label{invar-mea}
\bar{b}(x)=\int_{V}b(x,z)\mu^x(\mathrm{d} z),\qquad x \in V,\end{aligned}$$ and $\mu^x$ is the unique invariant measure on $V$ of the transition semigroups for the following frozen equation: $$\begin{aligned}
\label{invariant}
\mathrm{d} Y^{x}_t =\{AY^{x}_t+F(x,Y^{x}_t)\}\mathrm{d}t+G(x,Y^{x}_t)\mathrm{d}{W}_t, \qquad Y^{x}_0=y \in V.\end{aligned}$$

According to the definition of $\bar b$ (\[invar-mea\]) and conditions [(A1)-(A3)]{}, it is easy to prove $\bar b$ also satisfies the Lipschitz and growth conditions. Here, we omit the proof. Then, we have the following lemma.

Suppose that conditions [(A1)-(A3)]{} are satisfied. Then, for any initial value $X_{0} \in V_\beta,\beta>\alpha$, 