# Gradient flows and Ricci curvature for finite Markov chains

### (Univ. Bonn)

**Date:** Feb 15, 2012,
**time:** 16:30

**Place:** Aula Dini (SNS)

**Abstract.** Since the seminal work of Jordan, Kinderlehrer and Otto, it is known

that the heat flow on $R^n$ can be regarded as the gradient flow of

the entropy in the Wasserstein space of probability measures.

Meanwhile this interpretation has been extended to very general

classes of metric measure spaces, but it seems to break down if the

underlying space is discrete.

In this talk we shall present a new metric on the space of probability

measures on a discrete space, based on a discrete Benamou-Brenier

formula. This metric defines a Riemannian structure on the space of

probability measures and it allows to prove a discrete version of the

JKO-theorem.

This naturally leads to a notion of Ricci curvature based on convexity

of the entropy in the spirit of Lott-Sturm-Villani. We shall discuss

how this is related to functional inequalities and present discrete

analogues of results from Bakry-Emery and Otto-Villani.

This is partly joint work with Matthias Erbar (Bonn).