Stochastic maximal regularity for rough time-dependent problems

TL;DR

This paper develops a unified framework for stochastic maximal regularity in time-dependent PDEs with noise, extending existing theories to rough coefficients and providing optimal regularity results in various function spaces.

Contribution

It extends stochastic maximal regularity theory to rough, time-dependent coefficients and unifies semigroup and PDE approaches for a broad class of parabolic problems.

Findings

Established $L^{p}(L^{q})$ estimates for high-order systems with VMO regularity.

Derived $L^{p}(L^{p})$ estimates for second order systems with continuous coefficients.

Obtained tent space estimates for divergence form equations with measurable coefficients.

Abstract

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For $2m$-th order systems with $VMO$ regularity in space, we obtain $L^{p}(L^{q})$ estimates for all $p>2$ and $q\geq 2$, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain $L^{p}(L^{p})$ estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces $T^{p,2}_{\sigma}$ of Coifman-Meyer-Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.