Stochastic maximal L^p(L^q)-regularity for second order systems with periodic boundary conditions

In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω), and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove L^p((0,T)× Ω,t^κ dt;H^σ,q(T^d))-estimates. The main novelty is that we do not require p = q. Moreover, we allow arbitrary σ ∈ R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.