On the geometry of geodesics in discrete optimal transport

TL;DR

This paper investigates the structure of geodesics in the space of probability measures on a discrete set under optimal transport, proving conditions for geodesics to remain supported within a subset, with implications for discrete Hamilton-Jacobi equations.

Contribution

It introduces a geometric condition ensuring the existence of support-preserving geodesics in discrete optimal transport spaces, and provides an extension result for subsolutions to discrete Hamilton-Jacobi equations.

Findings

Proves the existence of support-preserving geodesics under a geometric condition.

Establishes an extension result for discrete Hamilton-Jacobi subsolutions.

Provides insights into the geometry of discrete optimal transport spaces.

Abstract

We consider the space of probability measures on a discrete set $X$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \subseteq X$, it is natural to ask whether they can be connected by a constant speed geodesic with support in $Y$ at all times. Our main result answers this question affirmatively, under a suitable geometric condition on $Y$ introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.