Semilinear stochastic integral equations in L p

Wolfgang Desch*, Stig Olof Londen

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

2 Citations (Scopus)

Abstract

We consider a semilinear parabolic stochastic integral equation (Formula presented.) Here t ∈ [0, T], ω in a probability space Ω,x In a σ-finite measure space B with (positive) measure Λ. The kernels aμ(t) are multiples of tμ-1. The operator A: D(A) ⊂ Lp(B) → Lp(B) is such that (–A)is a nonnegative operator. The convolution integrals aβ ⁎ Gk are stochastic convolutions with respect to independent scalar Wiener processes Wk. F: [0, T] × Ω × D((-A)θ) → Lp(B) and G: [0, T]×Ω×D((-A)θ) → Lp(B, l2) are nonlinear with suitable Lipschitz conditions. We establish an Lp-theory for this equation, including existence and uniqueness of solutions, and regularity results in terms of fractional powers of (-A) and fractional derivatives in time.

Original languageEnglish
Title of host publicationProgress in Nonlinear Differential Equations and Their Application
PublisherSpringer US
Pages131-166
Number of pages36
Volume80
DOIs
Publication statusPublished - 2011
MoE publication typeA3 Part of a book or another research book

Publication series

NameProgress in Nonlinear Differential Equations and Their Application
Volume80
ISSN (Print)1421-1750
ISSN (Electronic)2374-0280

Keywords

  • Nonnegative operator
  • Regularity
  • Semilinear stochastic integral equations
  • Singular kernel
  • Stochastic fractional differential equation
  • Volterra equation

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