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in Probabilistic approach to geometry, Adv. Stud. Pure Math. 57, Math. Soc. Japan (2010), 343–381.
This is a gentle introduction to the context and results of my article Ricci curvature of Markov chains on metric spaces. It begins with a description of classical Ricci curvature in Riemannian geometry, as well as a reminder for discrete Markov chains. A number of open problems are mentioned.