Quantitative stability for Bakry--\'Emery log-Sobolev and Talagrand inequalities

Abstract

We establish quantitative $L^1$-stability estimates for the Bakry--\'Emery log-Sobolev and Talagrand inequalities with a universal exponent of 1/19 governing the corresponding deficits. Our approach relies on a Maurey-type argument combined with stability estimates for the Pr\'ekopa--Leindler inequality. In the radial setting, the exponent of both deficits can be improved to $1/2,$ which turns out to be optimal. As an application, we establish an estimate for the hypercontractivity deficit of the Hopf--Lax semigroup in the Bakry--\'Emery setting. In particular, these stability results provide elementary characterizations for the equality cases in the previously mentioned inequalities.