A McKean-Pontrygin maximum principle for entropic-regularized optimal transport

TL;DR

This paper introduces a mean-field approach to dynamic optimal transport using a McKean-Pontryagin maximum principle, avoiding stochastic path sampling and unifying deterministic and stochastic problems.

Contribution

It presents a fully variational methodology with constrained Hamiltonian equations, connecting to existing stochastic differential equation formulations.

Findings

Avoids sampling over stochastic paths

Provides a unified treatment of deterministic and stochastic transport

Connects to forward-backward stochastic differential equations

Abstract

This note outlines a mean-field approach to dynamic optimal transport problems based on the recently proposed McKean-Pontryagin maximum principle. Key aspects of the proposed methodology include i) avoidance of sampling over stochastic paths, ii) a fully variational approach leading to constrained Hamiltonian equations of motion, and iii) a unified treatment of deterministic and stochastic optimal transport problems. We also discuss connections to well-known dynamic formulations in terms of forward-backward stochastic differential equations and extensions beyond classical entropic-regularized transport problems.