  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.
