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7 - Ricci curvature, entropy, and optimal transport

from PART 1 - SHORT COURSES

Published online by Cambridge University Press:  05 August 2014

By
Shin-Ichi Ohta
Edited by
Yann Ollivier ,
Hervé Pajot  and
Cedric Villani
Shin-Ichi Ohta
Affiliation:
Kyoto University
Yann Ollivier
Affiliation:
Université de Paris XI
Hervé Pajot
Affiliation:
Université de Grenoble
Cedric Villani
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
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Summary

Abstract

This chapter comprises the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure spaces satisfying this convexity condition.

Introduction

This chapter is extended notes based on the author's lecture series at the summer school at Université Joseph Fourier, Grenoble: “Optimal Transportation: Theory and Applications.” The aim of these five lectures (corresponding to Sections 7.3–7.7) was to review the recent impressive development on the interplay between optimal transport theory and Riemannian geometry. Ricci curvature and entropy are the key ingredients. See [Lo2] for a survey in the same spirit with a slightly different selection of topics.

Optimal transport theory is concerned with the behavior of transport between two probability measures in a metric space. We say that such transport is optimal if it minimizes a certain cost function typically defined from the distance of the metric space. Optimal transport naturally inherits the geometric structure of the underlying space; in particular Ricci curvature plays a crucial role for describing optimal transport in Riemannian manifolds. In fact, optimal transport is always performed along geodesics, and we obtain Jacobi fields as their variational vector fields. The behavior of these Jacobi fields is controlled by the Ricci curvature as is usual in comparison geometry. In this way, a lower Ricci curvature bound turns out to be equivalent to a certain convexity property of entropy in the space of probability measures.

Type
Chapter
Information
Optimal Transport
Theory and Applications
 , pp. 145 - 200
Publisher: Cambridge University Press
Print publication year: 2014

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