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).http://cvgmt.sns.it/seminar/287/
