Logarithmic Sobolev inequality for diffusion semigroups

TL;DR

This paper discusses the derivation of logarithmic Sobolev inequalities for diffusion semigroups, focusing on the Ornstein-Uhlenbeck process, using both the Bakry-Emery criterion and optimal mass transportation methods.

Contribution

It introduces an alternative approach using optimal mass transportation to establish logarithmic Sobolev inequalities, complementing the traditional Bakry-Emery criterion.

Findings

Bakry-Emery criterion effectively derives functional inequalities.

Optimal mass transportation provides a new method for logarithmic Sobolev inequalities.

Application to Ornstein-Uhlenbeck semigroup demonstrates the methods' effectiveness.

Abstract

Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincar\'e or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequality.