The Annals of Probability

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

François Bolley, Ivan Gentil, and Arnaud Guillin

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Abstract

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.

Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 261-301.

Dates
Received: October 2015
Revised: March 2017
First available in Project Euclid: 5 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1517821223

Digital Object Identifier
doi:10.1214/17-AOP1184

Mathematical Reviews number (MathSciNet)
MR3758731

Zentralblatt MATH identifier
06865123

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60E15: Inequalities; stochastic orderings

Keywords
Logarithmic Sobolev inequality Talagrand inequality Brascamp–Lieb inequality Fokker–Planck equations optimal transport

Citation

Bolley, François; Gentil, Ivan; Guillin, Arnaud. Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities. Ann. Probab. 46 (2018), no. 1, 261--301. doi:10.1214/17-AOP1184. https://projecteuclid.org/euclid.aop/1517821223


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