Gradient flows of the entropy for finite Markov chains

Details Gradient flows of the entropy for finite Markov chains Gradient flows of the entropy for finite Markov chains

2011-02-25

arxiv.org/abs/1102.5238v2

Mathematics

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An error in Example 2.6 has been corrected and several changes have been made accordingly. To appear in J. Funct. Anal

Scientific paper

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.

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