Quantitative logarithmic Sobolev inequalities and stability estimates

Max  Fathi; Emanuel  Indrei; Michel  Ledoux

doi:10.3934/dcds.2016097

Article Contents

2016, Volume 36, Issue 12: 6835-6853. Doi: 10.3934/dcds.2016097

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Quantitative logarithmic Sobolev inequalities and stability estimates

1.
Université Pierre et Marie Curie, Paris
2.
Carnegie Mellon University, Pittsburgh
3.
University of Toulouse and Institut Universitaire de France, Toulouse

Received: December 2015

Revised: June 2016

Published: May 2017

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Abstract

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincaré 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 ${ L}^1$-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-Émery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.

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Mathematics Subject Classification: Primary: 58J35, 60J60.

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