The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction

TL;DR

This paper establishes the convexity of the first eigenvalue of the Ornstein-Uhlenbeck operator under Minkowski addition and proves the log-concavity of the eigenfunction for convex domains, extending classical inequalities to this setting.

Contribution

It proves a Brunn-Minkowski type inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and characterizes equality cases, also showing eigenfunction log-concavity for convex domains.

Findings

Eigenvalue is convex under Minkowski addition.

Eigenfunction is log-concave in convex domains.

Characterization of equality cases in convex sets.

Abstract

We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize the equality cases in some classes of convex sets. We also prove that the corresponding (positive) eigenfunction is log-concave if the domain is convex.