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

TL;DR

This paper introduces dimension-dependent refinements of key functional inequalities using optimal transport and Borell-Brascamp-Lieb methods, leading to improved concentration, contraction, and convergence results for stochastic processes.

Contribution

It provides novel dimensional improvements of the logarithmic Sobolev, Talagrand, and Brascamp-Lieb inequalities with applications to stochastic differential equations.

Findings

Dimensionally sharpened inequalities with optimal scale behavior

Enhanced concentration and contraction bounds

Improved understanding of stochastic differential equations' solutions

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 behaviour for the laws of solutions to stochastic differential equations.