Invariant Measures and Maximal L^2 Regularity for Nonautonomous Ornstein-Uhlenbeck Equations

TL;DR

This paper characterizes the domain of a linear parabolic operator involving nonautonomous Ornstein-Uhlenbeck operators in L^2 spaces and establishes optimal regularity results for related evolution equations.

Contribution

It provides a new characterization of the domain of the operator in invariant L^2 spaces and derives optimal L^2 regularity results for time-dependent Ornstein-Uhlenbeck evolution equations.

Findings

Characterization of the domain of the operator in invariant L^2 spaces.

Establishment of optimal L^2 regularity for evolution equations.

Identification of invariant measures for the associated semigroup.

Abstract

We characterize the domain of the realization of the linear parabolic operator Gu := u_t + L(t)u (where, for each real t, L(t) is an Ornstein-Uhlenbeck operator), in L^2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L^2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators.