Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains

Abstract

We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains 𝒪⊆ ℝ ^d in the scale , 1/τ=α/d+1/p, p≥2 fixed. The Besov smoothness in this scale determines the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. The proofs are performed by establishing weighted Sobolev estimates and combining them with wavelet characterizations of Besov spaces.

Keywords:

• semilinear stochastic partial differential equation
• Besov space
• weighted Sobolev space
• nonlinear approximation
• wavelet expansion

2000 AMS Subject Classifications :

• 60H15
• Secondary: 46E35
• 65C30

Acknowledgements

This work has been supported by a doctoral scholarship of the Philipps-Universität Marburg and by Deutsche Forschungsgemeinschaft (DFG), grants DA 360/13-1, RI 599/4-1, SCHI 419/5-1. We thank Felix Lindner for his helpful comments on parts of this paper.