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

We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains O subset of R-d in the scale B-tau,tau(alpha) (O), 1/tau = alpha/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.