Gradient flows of the entropy for finite Markov chains

Abstract

Let K be an irreducible and reversible Markov kernel on a finite set X . We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in R n by Jordan, Kinderlehrer and Otto (1998). The metric W is similar to, but different from, the L 2 -Wasserstein metric, and is defined via a discrete variant of the Benamou–Brenier formula.