as the set of all distributions $u$ on ${\mathbb{R}}^d$ such that $(1-\Delta)^{\sigma/2}u\in L_2$. For $u\in H^\sigma_2$, we define $$\label{e:2.22}
\|u\|_{H^{\sigma}_2}:=\|(1-\Delta)^{\sigma/2}u\|_{L_2}
:=\|{\mathcal{F}}^{-1}[(1+|\xi|^2)^{\sigma/2}{\mathcal{F}}(u)(\xi)]\|_{L_2}.$$ Similarly for $\ell_2$-valued $g=(g^1, g^2, \dots)$, $$\label{e:2.22n}
\|g\|_{H^{\sigma}_2(\ell_2)}:=\||(1-\Delta)^{\sigma/2}g|_{\ell_2}\|_{L_2}
:=\||{\mathcal{F}}^{-1}[(1+|\xi|^2)^{\sigma/2}{\mathcal{F}}(g)(\xi)]|_{\ell_2}\|_{L_2}.$$

Let ${\mathcal{P}}$ be the predictable $\sigma$-field and ${\mathcal{P}}^{d{\mathbb{P}}\times dt}$ be the completion of ${\mathcal{P}}$ with respect to $d{\mathbb{P}}\times dt$. For each $\sigma \in {\mathbb{R}}$, define the Banach space $${\mathbb{H}}^{\sigma}_2(T):=L_2(\Omega\times [0,T],{\mathcal{P}}, H^{\sigma}_2).$$ That is, $u\in {\mathbb{H}}^{\sigma}_2(T)$ if $u$ is an $H^{\sigma}_2$-valued ${\mathcal{P}}^{d{\mathbb{P}}\times dt}$-measurable process defined on $\Omega \times [0,T]$ so that $$\|u\|_{{\mathbb{H}}^{\sigma}_2(T)}:= \left( \E\int^T_0\|u (t, \cdot )\|^2_{H^{\sigma}_2}\,dt\right)^{1/2}<\infty.$$ For an $\ell_2$-valued process $g=(g^1,g^2,...)$, we write $g\in {\mathbb{H}}^{\sigma}_2(T,\ell_2)$ if $g^k\in {\mathbb{H}}^{\sigma}_2(T)$ for every $k\geq 1$ and $$\|g\|_{{\mathbb{H}}^{\sigma}_2(T,\ell_2)}:=
\left(  \E \int^T_0\|g\|^2_{H^{\sigma}_2(\ell_2)}\,dt \right)^{1/2}<\infty.$$ Denote ${\mathbb{L}}_2(T)={\mathbb{H}}^0_2(T)$ and ${\mathbb{L}}_2(T,\ell_2)={\mathbb{H}}^0_2(T,\ell_2)$. Write $g\in {\mathbb{H}}^{\infty}_0(T,\ell_2)$ if $g^k=0$ for all sufficiently large $k$, and each $g^k$ is of the type $$g^k=\sum_{i=1}^nI_{(\tau_{i-1},\tau_i]}(t) g^{ik}(x)$$ where $\tau_i$ are bounded stopping times with respect to ${\mathcal{F}}_t$ and $g^{ik} \in C^{\infty}_0 ({\mathbb{R}}^d)$. It is known ([@Kr99 Theorem 3.10]) that ${\mathbb{H}}^{\infty}_0(T,\ell_2)$ is dense in ${\mathbb{H}}^{\sigma}_2(T,\ell_2)$ for any $\sigma$.

Finally we use $U^{\sigma}_2$ to denote the family of $H^{\sigma}_2({\mathbb{R}}^d)$-valued ${\mathcal{F}}_0$-measurable random variables $u_0$ having $$\|u_0\|_{U^{\sigma}_2}:=\left( \E  \|u_0\|^2_{H^{\sigma}_2}
\right)^{1/2}<\infty.$$

\[lemma 2\] Suppose $a>0$. [(i)]{} Let $h=(h^1,h^2,\cdots) \in L_2(\Omega\times [0,T],{\mathcal{P}}, \ell_2)$. Then the equality $$I^{a}(\sum_{k=1}^{\infty} \int^{\cdot}_0 h^k (s) dW^k_s)(t)=\sum_{k=1}^{\infty} (I^{a} 