Optimal transport between random measures

TL;DR

This paper investigates optimal transport couplings between equivariant random measures on Riemannian manifolds, establishing existence, uniqueness, and approximation of optimal maps, and defining a new metric on measure spaces.

Contribution

It proves the existence and uniqueness of optimal couplings induced by transportation maps for equivariant random measures, and introduces a metric on measure spaces based on optimal transport costs.

Findings

Unique optimal couplings are induced by transportation maps.

Optimal transportation maps can be approximated by classical solutions.

The transportation cost defines a metric on the space of measures.

Abstract

We study couplings $q^\bullet$ of two equivariant random measures $\lambda^\bullet$ and $\mu^\bullet$ on a Riemannian manifold $(M,d,m)$. Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and $\lambda^\omega\ll m$ we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, $q^\bullet=(id,T)_*\lambda^\bullet.$ We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of $L^p-$cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.