The Annals of Probability

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

Ildoo Kim, Kyeong-hun Kim, and Sungbin Lim

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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

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

Received: November 2016
Revised: March 2018
First available in Project Euclid: 4 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 45D05: Volterra integral equations [See also 34A12]

Stochastic partial differential equations time fractional derivatives maximal $L_{p}$-regularity multidimensional space–time white noise


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.

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