Since the seminal work of Jordan, Kinderlehrer and Otto, it is known that the heat flow on $\mathbb R^n$ 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).