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

Abstract

We present an $L_p$-theory ($p\ge 2$) for time-fractional stochastic partial differential equations driven by Lévy processes of the type $${\partial }_t^{\alpha }u=\sum _{i,j=1}^d{a}^{ij}{u}_{x^i{x}^j}+f+\sum _{k=1}^{\infty }{\partial }_t^{\beta }{\int }_0^t(\sum _{i=1}^d{\mu }^{ik}{u}_{x^i}+g^k)d{Z}_s^k$$ given with nonzero intial data. Here ${\partial }_t^{\alpha }$ and ${\partial }_t^{\beta }$ are the Caputo fractional derivatives, $\alpha \in (0,2),\beta \in (0,\alpha +1/p)$, and $\{Z_t^k:k=1,2,\cdots \}$ is a sequence of independent Lévy 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.

References in corpus (1)

• Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives