Stability of Poincar{\'e} constant

TL;DR

This paper investigates the stability of the sharp Poincaré constant for reversible diffusion processes, using spectral methods and Stein's technique, with applications to gamma distributions and log-concave measures.

Contribution

It introduces a spectral interpretation approach to analyze the stability of Poincaré constants, extending results to gamma and log-concave measures in one dimension.

Findings

Established stability results for gamma distributions.

Extended stability analysis to strictly log-concave measures.

Connected Poincaré stability with Brascamp-Lieb inequalities.

Abstract

We study stability of the sharp Poincar{\'e} constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincar{\'e} inequalities and Stein's method. In particular, these results are applied to the gamma distributions and to strictly log-concave measures in dimension one, giving stability for Brascamp-Lieb inequalities.