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

TL;DR

This paper establishes stochastic maximal regularity estimates for second order SPDEs with periodic boundary conditions, allowing flexible integrability and regularity parameters, which enhances analysis of nonlinear stochastic PDEs.

Contribution

It introduces a novel approach to stochastic maximal regularity without the restriction p=q, accommodating arbitrary regularity and weights, and develops new perturbation and multiplication results.

Findings

Proved stochastic maximal $L^p(L^q)$-regularity estimates for second order SPDEs.

Developed a general perturbation theory for SPDEs.

Established new results on pointwise multiplication in fractional Sobolev spaces.

Abstract

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 $(t,\omega)$, and H\"older continuous in space. Assuming stochastic parabolicity conditions, we prove $L^p((0,T)\times \Omega, t^{\kappa}\, \mathrm{d} t;H^{\sigma,q}(\mathbb{T}^d))$-estimates. The main novelty is that we do not require $p=q$. Moreover, we allow arbitrary $\sigma\in \mathbb{R}$ 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.