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

2025/12/16 by Xu, Zijian
#Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.14072

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.

Citations

• On the Monge-Kantorovich problem with additional linear constraints
• On the differential structure of metric measure spaces and applications
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Mass Transportation on Sub-Riemannian Manifolds
• Existence and uniqueness of optimal maps on Alexandrov spaces
• Polar factorization and monotone rearrangement of vector‐valued functions

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• Multi-marginal optimal transport: theory and applications