A Sobolev Space theory for stochastic partial differential equations with time-fractional derivatives

Abstract

In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t \sum_{k=1}^{\infty}\int^t_0 ( \Lambda^k(\omega,t,x)u+g^k(u))dw^k_t, $$ where $\alpha\in (0,2)$, $\beta <\alpha+\frac{1}{2}$, and $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ denote the Caputo derivative of order $\alpha$ and $\beta$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $\Omega$, $L$ is a second order operator of either divergence or non-divergence type, and $\Lambda^k$ are linear operators of order up to two. The coefficients of the equations depend on $\omega (\in \Omega), t,x$ and are allowed to be discontinuous. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.