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

Abstract

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $$ \partial_{t}^{\alpha}u=\left( \phi(\Delta) u +f(u) \right) + \partial_t^\beta \sum_{k=1}^\infty \int_0^t g^k(u)\,dw_s^k, \quad t>0, x\in \mathbb{R}^d; \,\,\, u(0,\cdot)=u_0 $$ as well as the SPDE driven by space-time white noise $$ \partial^{\alpha}_{t}u=\phi(\Delta)u + f(u) + \partial^{\beta-1}_{t}h(u) \dot{W}, \quad t>0,x\in \mathbb{R}^d; \quad u(0,\cdot)=u_{0}. $$ Here, $\alpha\in (0,1), \beta\in (-\infty, \alpha+1/2)$, $\{w_t^k : k=1,2,\cdots\}$ 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 $\partial_{t}^{\gamma}$ denotes the Caputo fractional derivative if $\gamma>0$ and the Riemann-Liouville fractional integral if $\gamma\leq0$. The the spatial non-local operator $\phi(\Delta)$ is a type of integro-differential operator whose symbol is $-\phi(|\xi|^2)$, where $\phi$ is a Bernstein function satisfying \begin{equation*} \kappa_0\left(\frac{R}{r}\right)^{\delta_{0}} \leq \frac{\phi(R)}{\phi(r)}, \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.