Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations

TL;DR

This paper establishes logarithmic Sobolev and Poincaré inequalities for nonautonomous parabolic equations, leading to hypercontractivity and asymptotic behavior results for the associated evolution operators.

Contribution

It introduces new inequalities and asymptotic analysis for nonautonomous Kolmogorov equations with unbounded coefficients, expanding understanding of their long-term behavior.

Findings

Proved logarithmic Sobolev inequalities for the evolution system of measures.

Established hypercontractivity of the evolution operator.

Analyzed asymptotic behavior of solutions over time.

Abstract

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincar\'e inequalities with respect to an associated evolution system of measures $\{\mu_t: t \in I\}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$.