We present a notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion is inspired by the seminal work of Lott, Sturm and Villani, who developed a synthetic notion of Ricci curvature for geodesic spaces based on convexity of the entropy along 2-Wasserstein geodesics.
In the discrete setting the role of the 2-Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy.
This approach allows us to obtain discrete analogues of results by Bakry–Emery and Otto–Villani.
This is joint work with Matthias Erbar (Bonn).