A Sobolev space theory for the stochastic partial differential equations with space-time non-local operators

We deal with the Sobolev space theory for stochastic partial differential equation (SPDE) driven by Wiener processes $$\begin{aligned} \partial _{t}^{\alpha }u=\left( \phi (\varDelta ) u +f(u) \right) + _t^\beta \sum _{k=1}^\infty \int _0^t g^k(u)\,dw_s^k, \quad t>0, \,x\in {\mathbb {R}}^d \end{aligned}$$ as well SPDE space-time white noise ^{\alpha }_{t}u=\phi )u f(u) ^{\beta -1}_{t}h(u) {\dot{W}}, t>0,\, x\in {R}}^d. Here, $$\alpha \in (0,1), \beta < \alpha +1/2$$ , $$\{w_t^k : k=1,2,\ldots \}$$ is a family of independent one-dimensional and $${\dot{W}}$$ defined on $$[0,\infty )\times {R}}^d$$ . The time non-local operator $$\partial }$$ denotes Caputo fractional derivative order $$ function $$\phi Bernstein function, spatial )$$ integro-differential whose symbol $$-\phi (|\xi |^2)$$ In other words, infinitesimal generator d-dimensional subordinate Brownian motion. prove uniqueness existence results in spaces obtain maximal regularity solutions.