3 - Logarithmic Sobolev inequality for diffusion semigroups

Summary

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 Poincaré or logarithmic Sobolev inequalities. Moreover, an alternative method using the optimal mass transportation is also given to obtain the logarithmic Sobolev inequality.

Introduction

The goal of this course (given in 2009 in Grenoble) is to introduce inequalities as Poincaré or logarithmic Sobolev for diffusion semigroups. We will focus more on examples than on the general theory. A main tool to obtain those inequalities is the so-called Bakry–Emery Γ₂-criterion. This criterion is well known to prove such inequalities and has also been used many times for other problems; see, for instance, [BÉ85, Bak06]. We will focus on the example of the Ornstein–Uhlenbeck semigroup and on the Γ₂-criterion.

In Section 3.2 we investigate the main example of the Ornstein–Uhlenbeck semigroup, whereas in Section 3.3 we show how the Γ₂-criterion implies such inequalities. In Section 3.4 we will explain an alternative method to get a logarithmic Sobolev inequality under curvature assumption. It is called the mass transportation method and has been introduced recently; see [CE02, OV00, CENV04, Vil09]. In this way we will also obtain another inequality called the Talagrand inequality or T₂inequality.

The Ornstein–Uhlenbeck semigroup and the Gaussian measure

In the general setting, if (X_t)_t≥0 is a Markov process on ℝⁿ, then the family of operators

P_t(f)(x) = E(f (X_t)),

where X₀ = x and a smooth function f, is defined as a Markov semigroup on ℝⁿ. There are two main examples. The first one is the heat semigroup, which is associated with the Brownian motion on ℝⁿ.