Dynamic characterization of barycentric optimal transport problems and their martingale relaxation

TL;DR

This paper extends the dynamic formulation of optimal transport to barycentric and martingale settings, establishing new links and analogues in weak optimal transport problems.

Contribution

It introduces a dynamic analogue for barycentric optimal transport and explores a martingale relaxation, connecting it to existing martingale Benamou-Brenier formulas.

Findings

Established a dynamic analogue for barycentric optimal transport.

Linked martingale relaxations to the martingale Benamou-Brenier formula.

Extended classical optimal transport concepts to new weak and martingale contexts.

Abstract

We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglb\"ock, Huesmann and K\"allblad.