Gradient flows of the entropy for finite Markov chains

TL;DR

This paper introduces a new metric on probability measures for finite Markov chains, demonstrating that their evolution can be viewed as a gradient flow of entropy, akin to the continuous heat flow interpretation.

Contribution

It constructs a novel metric W for finite Markov chains and proves the chain's evolution follows the gradient flow of entropy under this metric.

Findings

The metric W is similar to the L^2-Wasserstein metric but adapted for discrete spaces.

The Markov chain's law evolves as a gradient flow of entropy with respect to W.

The metric is defined via a discrete Benamou-Brenier formula.

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.