Stability of the Bakry-Émery theorem on [formula omitted

Abstract

We establish quantitative stability estimates for the Bakry-Émery bound on logarithmic Sobolev and Poincaré constants of uniformly log-concave measures. More specifically, we show that if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincaré constant as the standard Gaussian measure, then it almost splits off a Gaussian factor. Our results are dimension-free, leading to 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 near-extremals in the associated functional inequality.