Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

TL;DR

This paper investigates the spatial regularity of solutions to linear parabolic stochastic PDEs on Lipschitz domains using Besov spaces, which informs the convergence rates of nonlinear approximation schemes.

Contribution

It introduces a novel approach combining weighted Sobolev estimates and wavelet characterizations to analyze Besov regularity of stochastic PDE solutions on Lipschitz domains.

Findings

Besov regularity determines approximation convergence rates.

Weighted Sobolev estimates are effectively combined with wavelet methods.

Results apply to bounded Lipschitz domains in R^d.

Abstract

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.