a process $u$ such that $u^N=u$, a.s. on $A_{N,T}$ for each $N\geq 1$, hence, almost surely. Finally, the pathwise uniqueness follows in similar way. Thus, the proof is finished.

Fast-slow SPDEs Driven by fBm and Bm {#se-4}
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Throughout this section, we assume that the following conditions are fulfilled.

1.  The coefficients $b(x,y),F(x,y),G(x,y)$ of Eq. (\[spde-fs\]) are globally Lipschitz continuous in $x,y$, i.e., there exist two positive constants $C_{1},C_{2}$, such that $$\begin{aligned}
        |b(x_{1},y_{1})-b(x_{2},y_{2})|^{2}
        &\leq& C_{1}(|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2}),\cr
        |F(x_{1},y_{1})-F(x_{2},y_{2})|^{2}+|G(x_{1},y_{1})-G(x_{2},y_{2})|_{L_2(V)}^{2} &\leq& C_{2}(|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2}),\end{aligned}$$ for all $x_{1},x_{2},y_{1},y_{2}\in V$.

2.  The coefficients $b(x,y),F(x,y),G(x,y)$ of Eq. (\[spde-fs\]) satisfy linear growth conditions, i.e., there exist two positive constants $C_{3}, C_{4}$ such that $$\begin{aligned}
        |F(x,y)|^{2}+|G(x,y)|_{L_2(V)}^{2} &\leq& C_{3}(1+|x|^{2}+|y|^{2})\cr
        |b(x,y)|^{2} &\leq& C_{4}(1+|x|^{2}+|y|^{2}),\end{aligned}$$ for all $x,y\in V$.

3.  $g: V \rightarrow L(V)$ and $g^{\prime} : V \rightarrow L(V, L(V))$ are Lipschitz continuous in the senses of Eq. (\[g1\]) and Eq. (\[g2\]).

4.  There exist constants $\beta_{1}>0$ and $\beta_{2},\beta_{3}\in \mathbb{R}$ which are independent of $(x,y_{1},y_{2})$, such that $$\begin{aligned}
        \langle y_{1}, F(x,y_{1})\rangle & \leq& -\beta_{1} |y_{1}|^{2}+\beta_{2},\cr
        \langle y_{1}-y_{2}, F(x,y_{1})-F(x,y_{2}) \rangle &\leq & \beta_{3} |y_{1}-y_{2}|^{2},\end{aligned}$$ for all $x,y_{1},y_{2}\in V$.

5.  $\eta:=2\bar{\lambda}_1-2\beta_{3}-C_{2}>0,\kappa:=2\bar{\lambda}_1+2\beta_{1}-C_{3}>0$, where $\bar{\lambda}_1$ is the first eigenvalue of $-A$, $C_i, i=2,3$ and $\beta_i, i=1,2,3$ were given in (A1), (A2) and (A4).

\[remdissipative\] [Assumptions (A4) and (A5) 