Quantitative logarithmic Sobolev inequalities and stability estimates
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 -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.
Keywords
Cite
@article{arxiv.1410.6922,
title = {Quantitative logarithmic Sobolev inequalities and stability estimates},
author = {Max Fathi and Emanuel Indrei and Michel Ledoux},
journal= {arXiv preprint arXiv:1410.6922},
year = {2014}
}