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

TL;DR

This paper explores the geometric structure of a transportation metric on measures over finite Markov chains, revealing properties of geodesics, duality, and mass variation dynamics, with implications for related distances.

Contribution

It introduces a new dynamic transportation metric with a duality formula and analyzes geodesic non-locality and mass creation patterns in finite Markov chains.

Findings

Geodesics are supported on the entire state space almost everywhere.

Mass creation occurs early in optimal transport curves and decays over time.

The shift-transport distance is bounded above by the proposed metric.

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.