Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow

TL;DR

This paper introduces a Riemannian geometric framework for nonnegative measures on finite Markov chains, linking gradient flows to generalized heat equations and proving exponential convergence to equilibrium.

Contribution

It develops a discrete optimal transport geometry using Benamou--Brenier type metrics and characterizes the gradient flow of entropy as a generalized heat equation.

Findings

Identifies a Riemannian structure on measures over Markov chains.

Shows the gradient flow corresponds to a generalized heat equation with source.

Proves exponential convergence to equilibrium using a local ojasiewicz inequality.

Abstract

We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local \L{}ojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou--Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms.