Stochastic maximal $L^p$-regularity

TL;DR

This paper establishes a maximal $L^p$-regularity result for stochastic convolutions involving elliptic operators, extending classical inequalities to broader settings using advanced functional calculus and stochastic analysis techniques.

Contribution

It extends Krylov's $L^p(L^q)$-inequality for the Laplace operator to a wide class of elliptic operators on various domains.

Findings

Proves maximal $L^p$-regularity for stochastic convolutions with elliptic operators.

Extends classical inequalities to bounded domains with boundary conditions.

Provides maximal space--time $L^p$-regularity under invertibility assumptions.

Abstract

In this article we prove a maximal $L^p$-regularity result for stochastic convolutions, which extends Krylov's basic mixed $L^p(L^q)$-inequality for the Laplace operator on ${\mathbb{R}}^d$ to large classes of elliptic operators, both on ${\mathbb{R}}^d$ and on bounded domains in ${\mathbb{R}}^d$ with various boundary conditions. Our method of proof is based on McIntosh's $H^{\infty}$-functional calculus, $R$-boundedness techniques and sharp $L^p(L^q)$-square function estimates for stochastic integrals in $L^q$-spaces. Under an additional invertibility assumption on $A$, a maximal space--time $L^p$-regularity result is obtained as well.