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

TL;DR

This paper studies the regularity of solutions to stochastic parabolic equations on Lipschitz domains, establishing existence, uniqueness, and regularity results in weighted Sobolev and Besov spaces, with implications for numerical approximation.

Contribution

It introduces a framework for analyzing stochastic parabolic equations using weighted Sobolev and Besov spaces, providing new regularity results and embedding theorems.

Findings

Existence and uniqueness of solutions in weighted Sobolev spaces.

Hölder regularity in time for solutions.

Embedding of weighted Sobolev spaces into Besov spaces.

Abstract

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the H\"older regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^\alpha_{\tau,\tau}(O), 1/\tau=\alpha/d+1/p, \alpha > 0$. This leads to a H\"older-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.