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Publication 115

Quantitative Logarithmic Sobolev Inequalities and Stability Estimates

Authors:

M. Fathi
Université Pierre et Marie Curie
Paris, France


CMUE. Indrei
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh PA 15213-3890 USA


M. Ledoux
University of Toulouse
Toulouse, France
and
Institut Universitaire de France


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.
Get the paper in its entirety
14-CNA-027.pdf

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