At the end of the nineties, Jordan, Kinderlehrer, and Otto discovered
a new interpretation of the heat equation in R^n, as the gradient flow
of the entropy in the Wasserstein space of probability measures. In
this talk, I will present a discrete counterpart to this result: given
a reversible Markov kernel on a finite set, there exists a Riemannian
metric on the space of probability densities, for which the law of the
continuous time Markov chain is the gradient flow of the entropy.