## The Annals of Probability

### 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 semi-linear stochastic partial differential equations (SPDEs) of type \begin{equation*}\partial^{\alpha }_{t}u=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},\end{equation*} where $\alpha \in (0,2)$, $\beta <\alpha +\frac{1}{2}$ and $\partial^{\alpha }_{t}$ and $\partial^{\beta }_{t}$ denote the Caputo derivatives of order $\alpha$ and $\beta$, respectively. The processes $w^{k}_{t}$, $k\in \mathbb{N}=\{1,2,\ldots \}$, are independent one-dimensional Wiener processes, $L$ is either divergence or nondivergence-type second-order operator, and $\Lambda^{k}$ are linear operators of order up to two. 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.

We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal $L_{p}$-regularity of the solutions. By converting SPDEs driven by $d$-dimensional space–time white noise into the equations of above type, we also obtain an $L_{p}$-theory for SPDEs driven by space–time white noise if the space dimension $d<4-2(2\beta -1)\alpha^{-1}$. In particular, if $\beta <1/2+\alpha /4$ then we can handle space–time white noise driven SPDEs with space dimension $d=1,2,3$.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2087-2139.

Dates
Revised: March 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205704

Digital Object Identifier
doi:10.1214/18-AOP1303

Mathematical Reviews number (MathSciNet)
MR3980916

#### Citation

Kim, Ildoo; Kim, Kyeong-hun; Lim, Sungbin. A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives. Ann. Probab. 47 (2019), no. 4, 2087--2139. doi:10.1214/18-AOP1303. https://projecteuclid.org/euclid.aop/1562205704

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