Entropic Ricci Curvature on discrete Markov chains

Inspired by the famous work of Sturm and Lott-Villani to define the synthetic notion of lower Ricci curvature bound for metric measure spaces via the displacement convexity (in the sense of Optimal transport) of the Shannon entropy, in 2011 Erbar and Maas gave a modified definition of this Ricci curvature notion for discrete spaces (in particular, finite Markov chains). This curvature is known as entropic Ricci Curvature. After discussing the origin of the entropic Ricci curvature, I will talk about two equivalent reformulations of this curvature in terms of (1) Bochner’s formula and (2) gradient estimate with respect to the heat semigroup. These reformulations relate the entropic Ricci curvature to the Bakry-Émery curvature on weighted graphs.

11-25海报.pdf