Stochastic maximal Lp-regularity

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 ℝ^d to large classes of elliptic operators, both on ℝ^d and on bounded domains in ℝ^d with various boundary conditions. Our method of proof is based on McIntosh’s H^∞-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.