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

Qifan Mao, Xinyu Wang, Xiaoping Xue

TL;DR

We address nonconservative mass transport on graphs by developing a Benamou–Brenier–type metric $\mathcal{W}^{a,b}_p$ for nonnegative measures on finite reversible Markov chains. The metric couples a transport term with a mass-variation term constrained along a fixed direction $p$, enabling a continuum-like calculus on graphs. We show that the entropy is gradient-flowed by this metric, yielding a generalized heat equation with a source and proving exponential convergence to equilibrium via a Łojasiewicz inequality. This provides a coherent geometric framework for nonconservative diffusion on graphs and opens the door to duality and geodesic analyses in future work.

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 Łojasiewicz 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.

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

TL;DR

We address nonconservative mass transport on graphs by developing a Benamou–Brenier–type metric for nonnegative measures on finite reversible Markov chains. The metric couples a transport term with a mass-variation term constrained along a fixed direction , enabling a continuum-like calculus on graphs. We show that the entropy is gradient-flowed by this metric, yielding a generalized heat equation with a source and proving exponential convergence to equilibrium via a Łojasiewicz inequality. This provides a coherent geometric framework for nonconservative diffusion on graphs and opens the door to duality and geodesic analyses in future work.

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 Łojasiewicz 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.
Paper Structure (16 sections, 16 theorems, 203 equations)

This paper contains 16 sections, 16 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Basic settings of the Markov chain
  3. Metric space $(\mathcal{M},\mathcal{W}^{a,b}_p)$
  4. Definition of $\mathcal{W}^{a,b}_p$
  5. Solvability criteria for the continuity equation
  6. Metric axioms and completeness
  7. Gradient flows of the entropy functional
  8. Riemannian structures
  9. Entropy gradient flows and long-time asymptotics
  10. Conclusions
  11. Technical proofs
  12. Proof of Lemma \ref{['lem_ELinSq_potential_rewrite']}
  13. Proof of Lemma \ref{['lem_l1_bound']}
  14. Proof of Lemma \ref{['lem_positive_admissible_set']}
  15. Proof of Lemma \ref{['lem_differentiation_maximum']}
  16. ...and 1 more sections

Key Result

Lemma 3.2

For $\mu\in \mathcal{M}$ we have

Theorems & Definitions (27)

  • Definition 3.1: Benamou--Brenier-type metric on $\mathcal{M}$
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • Lemma 3.7
  • proof
  • ...and 17 more