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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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

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

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

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Zentralblatt MATH identifier

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

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


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.

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