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

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 Sobole 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 H o lder regularity in time is analysed. Moreover, we prove a general embedding of weighte $L_p(\mathcal{O})$ - Sobolev spaces into the scale o Besov spaces $B^\alpha_{\tau,\tau}(\mathcal{O})$ , $1/\tau=\alpha/d+1/p$ , $\alpha>0$ . This leads to a H o lder- 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.

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