Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics

TL;DR

This paper establishes the equivalence of Benamou-Brenier and Kantorovich formulations of optimal transport on certain sub-Riemannian manifolds, proving existence of minimizers and linking them to transport plans.

Contribution

It extends optimal transport theory to sub-Riemannian manifolds without abnormal geodesics, proving equivalence and existence results.

Findings

Equivalence of Benamou-Brenier and Kantorovich formulations on specified manifolds

Existence of minimizers for the Benamou-Brenier problem

Connection between minimizers and optimal transport plans

Abstract

We prove that the Benamou-Brenier formulation of the Optimal Transport problem and the Kantorovich formulation are equivalent on a sub-Riemannian connected and complete manifold $M$ without boundary and with no non-trivial abnormal geodesics, when the problems are considered between two measures with finite $2$-momentum. Furthermore, we prove the existence of a minimizer for the Benamou-Brenier formulation and link it to the optimal transport plan.