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

[1] Ahmad, N.; Kim, H. K.; McCann, R. J. Optimal transportation, topology and uniqueness, Bull. Sci. Math., Volume 1 (2011) no. 1, pp. 13-32 | DOI | MR | Zbl

[2] Beneš, V.; Štěpán, J. The support of extremal probability measures with given marginals, Mathematical statistics and probability theory, Vol. A (Bad Tatzmannsdorf, 1986), Reidel, Dordrecht, 1987, pp. 33-41 | DOI | Zbl

[3] Bernard, P.; Buffoni, B. Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), Volume 9 (2007) no. 1, pp. 85-121 | DOI | MR | Zbl

[4] Bernard, P.; Contreras, G. A generic property of families of Lagrangian systems, Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 1099-1108 | DOI | MR | Zbl

[5] Bianchini, S.; Caravenna, L. On the extremality, uniqueness and optimality of transference plans, Bull. Inst. Math. Acad. Sinica, Volume 4 (2009) no. 4, pp. 353-454 | MR | Zbl

[6] Chiappori, P.-A.; McCann, R. J.; Nesheim, L. P. Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness, Econom. Theory, Volume 42 (2010) no. 2, pp. 317-354 | DOI | MR | Zbl

[7] Clarke, F. H. Generalized gradients and applications, Trans. Amer. Math. Soc., Volume 205 (1975), pp. 247-262 | DOI | MR | Zbl

[8] Clarke, F. H. Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, J. Wiley & Sons, Inc., New York, 1983, xiii+308 pages | Zbl

[9] Federer, H. Geometric measure theory, Grundlehren Math. Wiss., 153, Springer-Verlag New York Inc., New York, 1969, xiv+676 pages | MR | Zbl

[10] Gangbo, W. Quelques problèmes d’analyse non convexe (1995) (Habilitation thesis, Université de Metz)

[11] Gangbo, W.; McCann, R. J. The geometry of optimal transportation, Acta Math., Volume 177 (1996) no. 2, pp. 113-161 | DOI | MR | Zbl

[12] Gangbo, W.; McCann, R. J. Shape recognition via Wasserstein distance, Quart. Appl. Math., Volume 58 (2000) no. 4, pp. 705-737 | DOI | MR | Zbl

[13] Gigli, N. On the inverse implication of Brenier-McCann theorems and the structure of $(P_2\left(M\right),W_2)$, Methods Appl. Anal., Volume 18 (2011) no. 2, pp. 127-158 | DOI | MR | Zbl

[14] Golubitsky, M.; Guillemin, V. Stable mappings and their singularities, Graduate Texts in Math., 14, Springer-Verlag, New York-Heidelberg, 1973, x+209 pages | MR | Zbl

[15] Hestir, K.; Williams, S. C. Supports of doubly stochastic measures, Bernoulli, Volume 1 (1995) no. 3, pp. 217-243 | DOI | MR | Zbl

[16] Levin, V. L. Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-Valued Anal., Volume 7 (1999) no. 1, pp. 7-32 | DOI | MR | Zbl

[17] Levin, V. L. On the generic uniqueness of an optimal solution in an infinite-dimensional linear programming problem, Dokl. Akad. Nauk, Volume 421 (2008) no. 1, pp. 21-23 | DOI | MR | Zbl

[18] Mañé, R. Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, Volume 9 (1996) no. 2, pp. 273-310 | DOI | MR | Zbl

[19] McCann, R. J. Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., Volume 11 (2001) no. 3, pp. 589-608 | DOI | MR | Zbl

[20] Milnor, J. W. Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, N.J., 1997, xii+64 pages (Revised reprint of the 1965 original)

[21] Moameni, A. Supports of extremal doubly stochastic measures (2014) (preprint)

[22] Rifford, L. Sub-Riemannian geometry and optimal transport, Springer Briefs in Mathematics, Springer, Cham, 2014, viii+140 pages | DOI | Zbl

[23] Srivastava, S. M. A course on Borel sets, Graduate Texts in Math., 180, Springer-Verlag, New York, 1998, xvi+261 pages | DOI | MR | Zbl

[24] Whitney, H. Geometric integration theory, Princeton University Press, Princeton, N.J., 1957, xv+387 pages | DOI | Zbl

[25] Zajíček, L. On the differentiability of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J., Volume 29 (1979) no. 3, pp. 340-348