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.

Maximal regularity for stochastic integral equations

“…Let (Ω, A, P) be a probability space with the filtration F = (F t ) t≥0 , and take W H (t), t ≥ 0, to be a cylindrical F-Brownian motion on the separable Hilbert space H. The stochastic integral with respect to W H on the right side of (1.1) is an L q (O; R)-valued random variable as 1 2 defined, e.g., in [7,Sec. 2.1].…”

“…Our result in this paper relies strongly on the deep approach of the recent paper [7], concerned with maximal regularity of (1.1) in the differential equation case, i.e., α = β = 1. In fact, the major part of the proof of [7] can be adapted almost without changes to apply to (1.1).…”

“…In fact, the major part of the proof of [7] can be adapted almost without changes to apply to (1.1). The only necessary major alteration is the construction of new, different k j -functions (see Proposition 3.3 below).…”

“…The only necessary major alteration is the construction of new, different k j -functions (see Proposition 3.3 below). Note that our k jfunctions are constructed by a different approach and do not reduce to those of [7] in case α = β = 1.…”

“…(See Corollary 3.2 below). Therefore, no modification or extension of the Da Prato-Kwapien-Zabczyk factorization argument used in [7] to obtain Bessel potential space regularity is required here. Obviously, this simpler proof also applies in case α = β = 1.…”