Stability of the Bakry-Emery theorem on $\mathbb{R}^n$

TL;DR

This paper establishes improved stability estimates for the Bakry-Emery theorem related to Poincaré and logarithmic Sobolev inequalities for log-concave measures, with implications for Gaussian concentration and measure splitting.

Contribution

It provides sharper quantitative bounds on stability for log-concave measures and extends results on measure splitting and concentration, using Stein's method and optimal transport.

Findings

Enhanced stability bounds for Poincaré constants

Dimension-free stability estimates for Gaussian concentration

New results on measure splitting for log-concave measures

Abstract

We prove stability estimates for the Bakry-Emery bound on Poincar\'e and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting that if a $1$-uniformly log-concave measure has almost the same Poincar\'e constant as the standard Gaussian measure, then it almost splits off a Gaussian factor, and establish similar new results for logarithmic Sobolev inequalities. As a consequence, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are based on Stein's method, optimal transport, and an approximate integration by parts identity relating measures and approximate optimizers in the associated functional inequality.