A Sobolev space theory for the Stochastic Partial Differential Equations with space-time non-local operators

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $${\mathrm{\partial }}_t^{\alpha }u=(\varphi (\mathrm{\Delta })u+f(u))+{\mathrm{\partial }}_t^{\beta }\sum _{k=1}^{\mathrm{\infty }}{\int }_0^t{g}^k(u)d{w}_s^k,t>0,x\in {\mathbb{R}}^d;u(0,\cdot )=u_0$$ as well as the SPDE driven by space-time white noise $${\mathrm{\partial }}_t^{\alpha }u=\varphi (\mathrm{\Delta })u+f(u)+{\mathrm{\partial }}_t^{\beta -1}h(u)\dot{W},t>0,x\in {\mathbb{R}}^d;u(0,\cdot )=u_0\mathrm{.}$$ Here, $\alpha \in (0,1),\beta \in (-\mathrm{\infty },\alpha +1\mathrm{/}2)$, ${w_t^k : k=1,2,\cdots}$ is a family of independent one-dimensional Wiener processes, and $\dot{W}$ is a space-time white noise defined on $[0,\infty)\times $\mathbb{R}$^d$\mathrm{.}Thetimenon-localoperator$\partial_{t}^{\gamma}$ denotes the Caputo fractional derivative if $\gamma >0$ and the Riemann-Liouville fractional integral if $\gamma \le 0$. The the spatial non-local operator $\varphi (\mathrm{\Delta })$ is a type of integro-differential operator whose symbol is $-\varphi (\mathrm{\mid }\xi {\mathrm{\mid }}^2)$, where $\varphi$ is a Bernstein function satisfying \begin{equation*} \kappa_0\left(\frac{R}{r}\right)^{\delta_{0}} \leq \frac{\phi®}{\phi®}, \qquad \forall,, 0<r<R<\infty \end{equation*} with some constants ${\kappa }_0>0$ and ${\delta }_0\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.