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Publication 14-CNA-027

Quantitative Logarithmic Sobolev Inequalities and Stability Estimates

M. Fathi
Université Pierre et Marie Curie
Paris, France
max.fathi@etu.upmc.fr

E. Indrei
Carnegie Mellon University
Pittsburgh PA 15213-3890 USA
egi@cmu.edu

M. Ledoux
University of Toulouse
Toulouse, France
and
Institut Universitaire de France
ledoux@math.univ-toulouse.fr

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 ${\rm L}^1$-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-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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