## The Annals of Probability

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

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

François Bolley, Ivan Gentil, and Arnaud Guillin

#### 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