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A survey of Ricci curvature for metric spaces and Markov chains. (English) Zbl 1204.53035

Kotani, Motoko (ed.) et al., Probabilistic approach to geometry. Proceedings of the 1st international conference, Kyoto, Japan, 28th July – 8th August, 2008. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-931469-58-7/hbk). Advanced Studies in Pure Mathematics 57, 343-381 (2010).
The text is a presentation of the general context of the author’s works [C. R. Math., Acad. Sci. Paris 345, No. 11, 643–646 (2007; Zbl 1132.53011) and J. Funct. Anal. 256, No. 3, 810–864 (2009; Zbl 1181.53015)], with a survey and comments on related work.
The goal is to generalize the notion of Ricci curvature for Riemannian manifolds to arbitrary metric spaces (such as graphs) and Markov chains. A new notion of Ricci curvature found by the author allows to obtain corresponding generalizations of a series of classical results in positive Ricci curvature: concentration of measure phenomena, logarithmic Sobolev inequalities, spectral gap estimates, etc.
The material is split into two parts. The first part gives the necessary background about concentration of measure, Ricci curvature of Riemannian manifolds, and convergence of Markov chains. Central for the second part is the notion of positive Ricci curvature. Informally speaking, in metric spaces with positive Ricci curvature, small balls are closer than their centers. The distance between balls is measured using the Kantorovich-Rubinstein transportation distance which is a special case of Wasserstein distances.
Special emphasis is put in the survey on open questions of varying difficulty.
For the entire collection see [Zbl 1190.60003].

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53B21 Methods of local Riemannian geometry
51F99 Metric geometry
60J05 Discrete-time Markov processes on general state spaces
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