On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

Petru A. Cioica, Kyeong Hun Kim, Kijung Lee, Felix Lindner

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains O ⊂ Rd with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces Hγ,q p, θ (O; T). The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the Hölder regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spacesBα τ, τ(O) 1/τ=α/d+1/p, α >0. This leads to a Hölder-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

Original languageEnglish
JournalElectronic Journal of Probability
Volume18
DOIs
Publication statusPublished - 2013 Sep 13

Fingerprint

Lipschitz Domains
Parabolic Equation
Stochastic Equations
Bounded Domain
Smoothness
Regularity
Nonlinear Approximation
Weighted Sobolev Spaces
Summability
Order of Convergence
Existence and Uniqueness of Solutions
Approximation Scheme
Sobolev Spaces
Dirichlet Boundary Conditions
Zero

Keywords

  • Adaptive numerical method
  • Besov space
  • Embedding theorem
  • Hölder regularity in time
  • L(L)-theory
  • Lipschitz domain
  • Nonlinear approximation
  • Quasi-banach space
  • Square root of Laplacian operator
  • Stochastic partial differential equation
  • Wavelet
  • Weighted sobolev space

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains. / Cioica, Petru A.; Kim, Kyeong Hun; Lee, Kijung; Lindner, Felix.

In: Electronic Journal of Probability, Vol. 18, 13.09.2013.

Research output: Contribution to journalArticle

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