The intrinsic dynamics of optimal transport

McCann, Robert J.; Rifford, Ludovic

Journal de l’École polytechnique - Mathématiques, Tome 3 (2016), pp. 67-98.

Nous nous intéressons aux coûts pour lesquels la solution du problème de transport optimal de Monge-Kantorovitch entre deux mesures de probabilités est unique. À l’heure actuelle, les seuls exemples connus de tels coûts lisses sur des variétés compactes nécessitent que l’une des variétés soit homéomorphe à une sphère. Nous introduisons une dynamique (multivaluée) associée au coût et exhibons des propriétés suffisantes pour l’unicité d’un plan de transport optimal. Cette approche nous permet de construire des coûts lisses sur des variétés compactes quelconques pour lesquels l’unicité d’un plan de transport optimal est assurée.

The question of which costs admit unique optimizers in the Monge-Kantorovich problem of optimal transportation between arbitrary probability densities is investigated. For smooth costs and densities on compact manifolds, the only known examples for which the optimal solution is always unique require at least one of the two underlying spaces to be homeomorphic to a sphere. We introduce a (multivalued) dynamics which the transportation cost induces between the target and source space, for which the presence or absence of a sufficiently large set of periodic trajectories plays a role in determining whether or not optimal transport is necessarily unique. This insight allows us to construct smooth costs on a pair of compact manifolds with arbitrary topology, so that the optimal transportation between any pair of probability densities is unique.

Classification : 49Q20, 28A35
Keywords: Optimal transport, Monge-Kantorovitch problem, optimal transport map, optimal transport plan, numbered limb system, sufficient conditions for uniqueness
Mot clés : Transport optimal, problème de Monge-Kantorovitch, application de transport optimale, plan de transport optimal, conditions suffisantes pour l’unicité

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