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

TL;DR

The paper extends the Benamou–Brenier and Kantorovich equivalence to sub-Riemannian manifolds without boundary and with no non-trivial abnormal geodesics, for initial and final measures with finite $2$-momentum. It introduces a relaxation of the BB problem using Young measures on $I\times HM$, proves the relaxed problem is equivalent to the classical BB and to Kantorovich, and employs a superposition principle to relate dynamic and static formulations. A measurable geodesic selector $S:M\times M\to \mathrm{Geod}(M)$ is constructed (via Suslin set theory) to connect Kantorovich plans to BB minimizers. Additionally, BB minimizers decompose into generalized curves supported on SR geodesics, providing a geometric interpretation that mirrors the Euclidean and Riemannian cases. This work thus establishes a rigorous foundation for dynamic OT on sub-Riemannian spaces and addresses complications arising from abnormal geodesics.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 2.1: See Theorem 1.5 in amgi13
• Remark 2.2
• Remark 2.3
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Theorem 3.6