Advanced Studies in Pure Mathematics

A survey of Ricci curvature for metric spaces and Markov chains

Yann Ollivier

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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


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