\Delta p(T-s,x) \,ds
&=&\int^T_0 D^{1-\beta}_t  \Phi(s)  \left(\partial^{\beta}_t p(\cdot, x)\right) (T-s) \,ds  \nonumber\\
&=&  \int^T_0   \phi (s ) \left( D^{1-\beta}_t  \partial^{\beta}_t p(\cdot, x) \right) (s) \,ds \nonumber \\
&=&\int^T_0   \phi( s )  \frac{\partial}{\partial s} p(s,x) \,ds. \label{e:fnnew}\end{aligned}$$ Recall that $D^2_x q$, $D^2_x p$ and $\frac{\partial}{\partial s}p$ are locally integrable in ${\mathbb{R}}^d\setminus\{0\}$ uniformly on the support of $\phi(t)$. By the integration by parts in $x$, Lemma \[lemma list\](vi), Lemma \[L:2.6\], and the Fubini’s Theorem, $$\begin{aligned}
\int_{{\mathbb{R}}^d}\int^T_0 \phi(s)  \psi (x) \Delta q (s,x) \,dsdx  &=&\int^T_0\int_{{\mathbb{R}}^d} \phi(s) \Delta \psi (x)  q (s,x) \,dxds\\
&=&\int_{{\mathbb{R}}^d} \Delta \psi(x)\left( \int_0^T \phi (s) \, D^{1-\beta}_t p(s,x) \,ds \right)dx\\
&=&\int_{{\mathbb{R}}^d} \Delta \psi (x) \left( \int^T_0 D^{1-\beta}_t  \Phi (s)   p(T-s,x) \,ds \right)dx\\
&=&\int_{{\mathbb{R}}^d} \psi (x)  \left( \int^T_0  D^{1-\beta}_t  \Phi(s)   \Delta p(T-s,x) \,ds \right)dx \\
&=&\int_{{\mathbb{R}}^d}\int^T_0   \phi( s ) \psi (x) \frac{\partial}{\partial s} p(s,x) \,dsdx.\end{aligned}$$ Since $\phi(t)$ and $\psi(x)$ are arbitrary, the lemma is proved.

**Key Estimates**
=================

In this section, we first define a stochastic Banach space and establish key lemmas. Then we study the $L_2$-theory of a model equation for SPDEs with fractional time-derivatives.

For $n=0,1,2,...$, define the Banach spaces $$H^n_2:=H^n_2({\mathbb{R}}^d) =\left\{u: u, D_xu, \cdots, D_x^n u \in L_2\right\}.$$ In general, for $\sigma \in {\mathbb{R}}$ define the space $H^{\sigma}_2=H^{\sigma}_2({\mathbb{R}}^d)=(1-\Delta)^{-\sigma/2}L_2$ 