On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

Abstract

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $\mathcal{O}\subset \mathbb{R}^d$ with both theoretical and numerical purpose. We use N.V. Krylov’s framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta} (\mathcal{O},T)$ 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 $L_p(\mathcal{O})$ -Sobolev spaces into the scale of Besov spaces $B^\alpha_{\tau,\tau}(\mathcal{O})$ $1/\tau=\alpha/d+1/p$, $\alpha>0$. 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.