Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Wasserstein geometry of nonnegative measures on finite Markov chains II: Geodesic and duality formulae

Qifan Mao, Xinyu Wang, Xiaoping Xue

TL;DR

The paper develops a Benamou--Brenier-type metric ${\mathcal{W}}^{a,b}_p$ for nonnegative measures on a finite reversible Markov chain, incorporating mass variation along a fixed direction ${p}$. It proves that ${\mathcal{W}}^{a,b}_p$ induces a geodesic metric space, establishes a strong nonlocality of geodesics (almost every time they are supported on all states), and derives a Kantorovich-type duality through Hamilton--Jacobi subsolutions, including a comparison bound with the shift--transport distance. The dual formulation uses a convex-analytic approach via Fenchel--Rockafellar duality, with a penalty term capturing the constrained source direction. These results provide an intrinsic geometric framework for nonconservative transport on graphs and suggest avenues for extensions to infinite graphs and applications to data-driven contexts.

Abstract

In this paper, we investigate the geodesic structure and the associated Kantorovich-type duality for a Benamou-Brenier-type transportation metric defined on the space of nonnegative measures over a finite reversible Markov chain. The metric is introduced through a dynamic formulation that combines transport and source costs along solutions of a nonconservative continuity equation, where mass variation is constrained to occur along a fixed strictly positive reference direction. We show that geodesics associated with this metric exhibit a non-locality property: almost every time, they are supported on the whole state space, independently of the choice of endpoints. Moreover, along optimal curves, the source term displays a characteristic temporal profile, with mass creation occurring at early times and subsequent decay as the curve approaches the target measure. As an application of this property, we compare our metric with the shift-transport distance and prove that the latter is always bounded above by our metric. Finally, we establish a Kantorovich-type duality formula in terms of Hamilton-Jacobi subsolutions, which provides a characterization of the metric and highlights the role of the momentum associated with geodesic curves.

Wasserstein geometry of nonnegative measures on finite Markov chains II: Geodesic and duality formulae

TL;DR

The paper develops a Benamou--Brenier-type metric for nonnegative measures on a finite reversible Markov chain, incorporating mass variation along a fixed direction . It proves that induces a geodesic metric space, establishes a strong nonlocality of geodesics (almost every time they are supported on all states), and derives a Kantorovich-type duality through Hamilton--Jacobi subsolutions, including a comparison bound with the shift--transport distance. The dual formulation uses a convex-analytic approach via Fenchel--Rockafellar duality, with a penalty term capturing the constrained source direction. These results provide an intrinsic geometric framework for nonconservative transport on graphs and suggest avenues for extensions to infinite graphs and applications to data-driven contexts.

Abstract

In this paper, we investigate the geodesic structure and the associated Kantorovich-type duality for a Benamou-Brenier-type transportation metric defined on the space of nonnegative measures over a finite reversible Markov chain. The metric is introduced through a dynamic formulation that combines transport and source costs along solutions of a nonconservative continuity equation, where mass variation is constrained to occur along a fixed strictly positive reference direction. We show that geodesics associated with this metric exhibit a non-locality property: almost every time, they are supported on the whole state space, independently of the choice of endpoints. Moreover, along optimal curves, the source term displays a characteristic temporal profile, with mass creation occurring at early times and subsequent decay as the curve approaches the target measure. As an application of this property, we compare our metric with the shift-transport distance and prove that the latter is always bounded above by our metric. Finally, we establish a Kantorovich-type duality formula in terms of Hamilton-Jacobi subsolutions, which provides a characterization of the metric and highlights the role of the momentum associated with geodesic curves.
Paper Structure (19 sections, 19 theorems, 334 equations, 2 figures)

This paper contains 19 sections, 19 theorems, 334 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notations
  4. Benamou--Brenier-type metric
  5. Geodesic geometry
  6. $(\mathcal{M},\mathcal{W}_p^{a,b})$ is a geodesic space
  7. Non-locality of geodesics
  8. Comparison with the shift--transport metric
  9. Conditioned uniqueness of geodesics
  10. Geodesic equation
  11. Duality formula with Hamilton--Jacobi subsolutions
  12. Conclusions
  13. Proofs in Section \ref{['sec_geodesic']}
  14. Proof of Lemma \ref{['lem_alpha_convex']}
  15. Proof of Lemma \ref{['lem_W_rewrite']}
  16. ...and 4 more sections

Key Result

Theorem 2.2

$(\mathcal{M},\mathcal{W}^{a,b}_p)$ is a metric space.

Figures (2)

  • Figure 1: Rays starting from $(0.6,0.8)^{\mathrm{T}}$
  • Figure 2: Rays starting from $(0.05,1)^{\mathrm{T}}$

Theorems & Definitions (36)

  • Definition 2.1: Benamou--Brenier-type metric on $\mathcal{M}$
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Corollary 3.5
  • proof
  • Lemma 3.6
  • ...and 26 more