Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations

TL;DR

This paper analyzes the long-term behavior and hypercontractivity of nonautonomous Ornstein-Uhlenbeck equations, providing insights into their asymptotic properties in both periodic and non-periodic contexts.

Contribution

It establishes the asymptotic behavior and hypercontractivity of evolution operators for nonautonomous Ornstein-Uhlenbeck equations, extending understanding beyond autonomous cases.

Findings

Demonstrates asymptotic stability of the evolution operator

Shows hypercontractivity of the associated evolution semigroup

Analyzes behavior in both periodic and non-periodic settings

Abstract

In this paper we investigate a class of nonautonomous linear parabolic problems with time depending Ornstein-Uhlenbeck operators. We study the asymptotic behavior of the associated evolution operator and evolution semigroup in the periodic and non-periodic situation. Moreover, we show that the associated evolution operator is hypercontractive.