Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow
Analysis of PDEs
2026-01-21 v1 Probability
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.
Keywords
Cite
@article{arxiv.2601.13073,
title = {Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow},
author = {Qifan Mao and Xinyu Wang and Xiaoping Xue},
journal= {arXiv preprint arXiv:2601.13073},
year = {2026}
}