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

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.

💡 Research Summary

This paper introduces a novel extension of the multi‑marginal optimal transport (MOT) problem called Hierarchical Jump Multi‑Marginal Optimal Transport (HJMO‑T). The key idea is to allow the transport plan to “jump” over selected intermediate spaces by augmenting each intermediate space (X_k) (for (1\le k\le K-1)) with an isolated point (\partial_k). The extended space (\bar X_k = X_k\cup{\partial_k}) retains the Polish topology, and the isolated point represents the decision to skip that stage. A path (\omega = (x_0,\dots,x_K)) in the product space (\Omega = \prod_{k=0}^K \bar X_k) is processed by an active‑index map (I(\omega)) that extracts the ordered list of actually visited stages, always including the initial and terminal indices. The cost of a path is then defined as the sum of pairwise costs (c_{i,j}) over consecutive active indices, i.e.
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