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

TL;DR

This paper develops an $L_p$-theory for time-fractional stochastic PDEs driven by Levy processes, establishing existence, uniqueness, and regularity of solutions in Sobolev spaces with fractional derivatives.

Contribution

It introduces a novel Sobolev space framework for analyzing time-fractional SPDEs with Levy noise, including maximal regularity results and handling random coefficients.

Findings

Proved existence and uniqueness of solutions in Sobolev spaces.

Established maximal regularity for solutions.

Extended the theory to equations with random coefficients.

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.