Optimal transportation with infinitely many marginals

TL;DR

This paper extends optimal transportation theory to infinitely many marginals, establishing existence, uniqueness, and characterization of solutions by linking it to Wasserstein barycenters, thus broadening the scope of multi-marginal optimal transport problems.

Contribution

It introduces a novel formulation of optimal transportation with infinitely many marginals and connects it to Wasserstein barycenters, extending prior finite-marginal results.

Findings

Proved existence and uniqueness of solutions.

Characterized the optimizer in the infinite-marginal case.

Established a relationship with Wasserstein barycenters.

Abstract

We formulate and study an optimal transportation problem with infinitely many marginals; this is a natural extension of the multi-marginal problem studied by Gangbo and Swiech (1998). We prove results on the existence, uniqueness and characterization of the optimizer, which are natural extensions of the results of Gangbo and Swiech. The proof relies on a relationship between this problem and the problem of finding barycenters in the Wasserstein space, a connection first observed for finitely many marginals by Agueh and Carlier (2011).