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 O ⊂ Rd with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces Hγ,q _{p, θ} (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(O)-Sobolev spaces into the scale of Besov spacesBα _{τ, τ}(O) 1/τ=α/d+1/p, α >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.