Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

François Bolley ^[1] ; Ivan Gentil ^[2] ; Arnaud Guillin ^[3]
1. [1] Pierre and Marie Curie University
  
  Pierre and Marie Curie University
  
  París, Francia
2. [2] University of Lyon System
  
  University of Lyon System
  
  Arrondissement de Lyon, Francia
3. [3] University of Clermont Auvergne
  
  University of Clermont Auvergne
  
  Arrondissement de Clermont-Ferrand, Francia
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Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 46, Nº. 1, 2018, págs. 261-301
Idioma: inglés
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Resumen
- In this work, we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities. For this, we use optimal transport methods and the Borell–Brascamp–Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behavior for the laws of solutions to stochastic differential equations.