Optimal Transportation and Functional Inequalities

Abstract

This chapter is a brief investigation of the links between optimal transportation methods and functional inequalities in the Markov operator framework of this monograph. After a brief introduction to the basic material on optimal transportation, the main topic of transportation cost inequalities and first examples for Gaussian measures are presented. Interpolation along the geodesics of optimal transport is used towards logarithmic Sobolev inequalities and transportation cost inequalities comparing relative entropy and Wasserstein distances between probability measures. An alternate approach to sharp Sobolev or Gagliardo–Nirenberg inequalities in Euclidean space is provided next along these lines. Non-linear Hamilton–Jacobi equations and hypercontractivity properties of their solutions, analogous to the ones for linear heat equations, are investigated in the further sections towards the relationships between (quadratic) transportation cost inequalities and logarithmic Sobolev inequalities. Contraction properties in Wasserstein space along with the heat semigroup are investigated in the Markov operator setting. The last section is a very brief overview of recent developments towards a notion of Ricci curvature lower bounds based on optimal transportation and the connection with the Γ-calculus developed in this work.