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

2025/07/28 by Citti, Giovanna, Galeotti, Mattia, Pinamonti, Andrea
#49Q22 #53C17 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2507.20959

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.

Citations

• Benamou-Brenier Formulation of Optimal Transport for Nonlinear Control Systems on Rd
• A relaxation viewpoint to Unbalanced Optimal Transport: duality, optimality and Monge formulation
• Three superposition principles: currents, continuity equations and curves of measures
• Continuity equations and ODE flows with non-smooth velocity
• Mass Transportation on Sub-Riemannian Manifolds
• Optimal Transportation under Nonholonomic Constraints
• Polar factorization and monotone rearrangement of vector‐valued functions
• A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

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