## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 343 - 381

### A survey of Ricci curvature for metric spaces and Markov chains

#### Abstract

This text is a presentation of the general context and results of [Oll07] and [Oll09], with comments on related work. The goal is to present a notion of Ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive Ricci curvature, such as spectral gap estimates, concentration of measure or log-Sobolev inequalities.

The necessary background (concentration of measure, curvature in Riemannian geometry, convergence of Markov chains) is covered in the first section. Special emphasis is put on open questions of varying difficulty.

#### Article information

**Source***Probabilistic Approach to Geometry*, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 343-381

**Dates**

Received: 14 December 2008

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543086328

**Digital Object Identifier**

doi:10.2969/aspm/05710343

**Mathematical Reviews number (MathSciNet)**

MR2648269

**Zentralblatt MATH identifier**

1204.53035

**Subjects**

Primary: 51F99: None of the above, but in this section 53B21: Methods of Riemannian geometry 60B99: None of the above, but in this section

**Keywords**

Ricci curvature concentration of measure Markov chains Wasserstein distances metric measure spaces

#### Citation

Ollivier, Yann. A survey of Ricci curvature for metric spaces and Markov chains. Probabilistic Approach to Geometry, 343--381, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710343. https://projecteuclid.org/euclid.aspm/1543086328