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 $\partial_{t}^αu=\left( ϕ(Δ) u +f(u) \right) + \partial_t^β\sum_{k=1}^\infty \int_0^t g^k(u)\,dw_s^k, \quad t>0, x\in \mathbb{R}^d; \,\,\, u(0,\cdot)=u_0$ as well as the SPDE driven by space-time white noise $\partial^α_{t}u=ϕ(Δ)u + f(u) + \partial^{β-1}_{t}h(u) \dot{W}, \quad t>0,x\in \mathbb{R}^d; \quad u(0,\cdot)=u_{0}.$ Here, $α\in (0,1), β\in (-\infty, α+1/2)$, $\{w_t^k : k=1,2,\cdots\}$ is a family of independent one-dimensional Wiener processes, and $\dot{W}$ is a space-time white noise defined on $[0,\infty)\times \mathbb{R}^d$. The time non-local operator $\partial_{t}^γ$ denotes the Caputo fractional derivative if $γ>0$ and the Riemann-Liouville fractional integral if $γ\leq0$. The the spatial non-local operator $ϕ(Δ)$ is a type of integro-differential operator whose symbol is $-ϕ(|ξ|^2)$, where $ϕ$ is a Bernstein function satisfying \begin{equation*} κ_0\left(\frac{R}{r}\right)^{δ_{0}} \leq \frac{ϕ(R)}{ϕ(r)}, \qquad \forall\,\, 0<r<R<\infty \end{equation*} with some constants $κ_0>0$ and $δ_0\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.

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