A Sobolev space theory for the time-fractional stochastic partial differential equations driven by Levy processes

Abstract

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by L\'evy processes of the type $$ \partial^{\alpha}_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f+\sum_{k=1}^{\infty}\partial^{\beta}_{t}\int_{0}^{t} (\sum_{i=1}^d\mu^{ik} u_{x^i} +g^k) dZ^k_{s} $$ given with nonzero intial data. Here $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ are the Caputo fractional derivatives, $\alpha\in (0,2), \beta\in (0,\alpha+1/p)$, and $\{Z^k_t:k=1,2,\cdots\}$ is a sequence of independent L\'evy processes. The coefficients are random functions depending on $(t,x)$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity of the solution.