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

Research output: Contribution to journalArticle

Abstract

In this article, we present an Lp-theory (p ≥ 2) for the semi-linear stochastic partial differential equations (SPDEs) of type ∂ t α u = L(ω,t,x)u +f (u)+∂ t β ∞ ∑ k=1 ∫ 0 t (∧k (ω,t,x)u +gk(u))dw t k, where α ∈ (0, 2), β < α + 1/2 and ∂ t α and ∂ t β denote the Caputo derivatives of order α and β, respectively. The processes w t k, k ∈ N = 1, 2, . . ., are independent one-dimensional Wiener processes, L is either divergence or nondivergence-type second-order operator, and ∧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 Lp-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 Lp-theory for SPDEs driven by space-time white noise if the space dimensiond < 4-2(2β -1-1. In particular, if β < 1/2 + α/4 then we can handle space-time white noise driven SPDEs with space dimension d = 1,2,3.

Original languageEnglish
Pages (from-to)2087-2139
Number of pages53
JournalAnnals of Probability
Volume47
Issue number4
DOIs
Publication statusPublished - 2019 Jan 1

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Stochastic Partial Differential Equations
Fractional Derivative
Sobolev Spaces
Space-time White Noise
Caputo Derivative
Linear partial differential equation
Wiener Process
Existence and Uniqueness Results
Strong Solution
Random Effects
Trapping
Semilinear
Linear Operator
Divergence
Regularity
Partial differential equations
Derivatives
Denote
Operator

Keywords

  • Maximal Lp-regularity
  • Multidimensional space-time white noise
  • Stochastic partial differential equations
  • Time fractional derivatives

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives. / Kim, Ildoo; Kim, Kyeong Hun; Lim, Sungbin.

In: Annals of Probability, Vol. 47, No. 4, 01.01.2019, p. 2087-2139.

Research output: Contribution to journalArticle

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