Quantitative logarithmic Sobolev inequalities and stability estimates

TL;DR

This paper improves classical logarithmic Sobolev inequalities for Gaussian measures under Poincaré conditions, providing quantitative stability estimates and bounds on deficits using Wasserstein distances, with applications in Bakry-Émery theory.

Contribution

It introduces enhanced logarithmic Sobolev inequalities with explicit deficit bounds in terms of Wasserstein distances, extending stability analysis under Poincaré conditions.

Findings

Lower bounds on the deficit in terms of quadratic Kantorovich-Wasserstein distance.

Lower bounds on the deficit in terms of L^1-Kantorovich-Wasserstein distance.

Applications to Bakry-Émery theory and the coherent state transform.

Abstract

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.