Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-dimensional Brownian motion $W_{\cdot}$. Let $\mathbb{D}^{s,2}$ denote the stochastic Sobolev space, that is the completion of the space of polynomial $\mathbb{R}^n$-valued $\mathcal{F}_1$-adapted random variables with respect to the norms $\|\cdot\|_{s,2}$ defined for any polynomial random variable $F$ as $$ \|F\|_{s,2}:=\|(I-L)^{s/2}F\|_{L^2(\Omega,\mathcal{F}_1)} $$ where $L$ is the infinitesimal generator of the Ornstein-Uhlenbeck semigroup on $L^2(\Omega,\mathcal{F}_1)$.
For any positive integer $N$, let $J_N$ be the orthogonal projection onto the first $N$-Wiener chaoses.
My question is: Does there exists a stochastic version of the Rellich–Kondrachov theorem. Specifically, if $\mathbb{D}^{s,2}$ relatively compact in $L^2(\Omega,\mathcal{F}_1)$ and are there known error rates for the projection $$ \sup_{F\in \mathbb{D}^{s,2};\,\|F\|_{s,2}\le 1}\, \|J_N(F)-F\|_{L^2(\Omega,\mathcal{F}_1)} \le ??? $$ for $s$ large enough?
