On the geometry of geodesics in discrete optimal transport

2018/05/15 by Erbar, Matthias, Maas, Jan, Wirth, Melchior · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1805.06040

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 ⊆ 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.

Cited by

• Fokker-Planck equations on discrete infinite graphs

Related

• Geodesics for a class of distances in the space of probability measures
• Transport type metrics on the space of probability measures involving singular base measures
• Principal Geodesic Analysis for Probability Measures under the Optimal\n Transport Metric
• Scaling limits of discrete optimal transport
• An Interpolating Distance between Optimal Transport and Fisher-Rao