Log-concavity of solutions of parabolic equations related to the Ornstein-Uhlenbeck operator and applications

TL;DR

This paper studies the log-concavity properties of solutions to parabolic equations associated with the Ornstein-Uhlenbeck operator, providing new proofs for related inequalities and eigenfunction properties.

Contribution

It offers a novel proof of log-concavity preservation and related inequalities for the Ornstein-Uhlenbeck operator's solutions.

Findings

Log-concavity of the kernel is established in convex domains.

Preservation of log-concavity for the flow is demonstrated.

A new proof of the Brunn-Minkowski inequality for the first eigenvalue is provided.

Abstract

In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of the log-concavity of the initial datum by the related flow. As an application, we give another proof of a Brunn-Minkowski type inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and of the log-concavity of the related first eigenfunction (both results have been proved in [9], by different methods).