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Article Dans Une Revue Discrete and Continuous Dynamical Systems - Series A Année : 2016

Quantitative logarithmic Sobolev inequalities and stability estimates

Résumé

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 ${\rm L}^1$-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-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.

Dates et versions

hal-01874822 , version 1 (14-09-2018)

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Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete and Continuous Dynamical Systems - Series A, 2016, 36 (12), pp.6835-6853. ⟨10.3934/dcds.2016097⟩. ⟨hal-01874822⟩
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