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arXiv · 1410.6922

Quantitative logarithmic Sobolev inequalities and stability estimates

Max Fathi·Emanuel Indrei·Michel Ledoux
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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 ${\rm L}^1$-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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Max Fathi, Emanuel Indrei, Michel Ledoux. 2014-10-25. Quantitative logarithmic Sobolev inequalities and stability estimates. https://arxiv.org/abs/1410.6922

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