Monge solutions and uniqueness in multi-marginal optimal transport with hierarchical jumps

TL;DR

This paper introduces Hierarchical Jump multi-marginal optimal transport (HJMOT), extending classical models to include jumps over intermediate spaces, ensuring solution existence and uniqueness under certain conditions, applicable to complex geometries.

Contribution

The paper develops a new HJMOT framework that guarantees existence and uniqueness of solutions, extending optimal transport theory to hierarchical jumps and complex geometries.

Findings

Existence of Kantorovich solutions in HJMOT

Uniqueness of Monge solutions under specific conditions

Framework applicable to Riemannian manifolds

Abstract

We introduce Hierarchical Jump multi-marginal transport (HJMOT), a generalization of multi-marginal optimal transport where mass can "jump" over intermediate spaces via augmented isolated points. Established on Polish spaces, the framework guarantees the existence of Kantorovich solutions and, under sequential differentiability and a twist condition, the existence and uniqueness of Monge solutions. This core theory extends robustly to diverse settings, including smooth Riemannian manifolds, demonstrating its versatility as a unified framework for optimal transport across complex geometries.