An optimal transportation principle for interacting paths and congestion

TL;DR

This paper introduces a modified optimal transport framework that accounts for path dependence and particle interactions, providing existence results, dual formulations, and extending classical theorems to this new setting.

Contribution

It develops a novel optimal transport model incorporating interaction effects and path dependence, extending Brenier's theorem and relating solutions to standard Monge-Kantorovich problems.

Findings

Proved existence of solutions under mild conditions.

Characterized minimizers via auxiliary Monge-Kantorovich problems.

Extended Brenier's theorem to interacting path transport.

Abstract

In this work we study a modification of the Monge-Kantorovich problem taking into account path dependence and interaction effects between particles. We prove existence of solutions under mild conditions on the data, and after imposing stronger conditions, we characterize the minimizers by relating them to an auxiliary Monge-Kantorovich problem of the more standard kind. With this notion of how particles interact and travel along paths, we produce a dual problem. The main novelty here is to incorporate an interaction effect to the optimal path transport problem. This covers for instance, $N$-body dynamics when the underlying measures are discrete. Lastly, our results include an extension of Brenier's theorem on optimal transport maps.