Existence and uniqueness of optimal transport maps

Cavalletti, Fabio; Huesmann, Martin

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 1367-1377 / Harvested from Numdam

Résumé

Let $(X, d, m)$ be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided $(X, d, m)$ satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.We also prove a stability property for Assumption 1: If $(X, d, m)$ satisfies Assumption 1 and $\tilde{m} = g \cdot m$ , for some continuous function $g > 0$ , then also $(X, d, \tilde{m})$ verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.

Publié le : 2015-01-01

MR 3425266 | Zbl 1331.49063

DOI : https://doi.org/10.1016/j.anihpc.2014.09.006

@article{AIHPC_2015__32_6_1367_0,
     author = {Cavalletti, Fabio and Huesmann, Martin},
     title = {Existence and uniqueness of optimal transport maps},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {32},
     year = {2015},
     pages = {1367-1377},
     doi = {10.1016/j.anihpc.2014.09.006},
     mrnumber = {3425266},
     zbl = {1331.49063},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2015__32_6_1367_0}
}

Cavalletti, Fabio; Huesmann, Martin. Existence and uniqueness of optimal transport maps. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) pp. 1367-1377. doi : 10.1016/j.anihpc.2014.09.006. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_6_1367_0/

Bibliographie

[1] L. Ambrosio, T. Rajala, Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces, Ann. Mat. Pura Appl. (2011), 1 -17 | MR 3158838

[2] L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 no. 2 (2004), 261 -301 | MR 2035027 | Zbl 1076.49023

[3] J. Bertrand, Existence and uniqueness of optimal maps on Alexandrov spaces, Adv. Math. 219 no. 3 (2008), 838 -851 | MR 2442054 | Zbl 1149.49002

[4] S. Bianchini, F. Cavalletti, The Monge problem for distance cost in geodesic spaces, Commun. Math. Phys. 318 (2013), 615 -673 | MR 3027581 | Zbl 1275.49080

[5] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Commun. Pure Appl. Math. 44 no. 4 (1991), 375 -417 | MR 1100809 | Zbl 0738.46011

[6] F. Cavalletti, Monge problem in metric measure spaces with Riemannian curvature-dimension condition, Nonlinear Anal. 99 (2014), 136 -151 | MR 3160530 | Zbl 1283.49058

[7] N. Gigli, Optimal maps in non branching spaces with Ricci curvature bounded from below, Geom. Funct. Anal. (2011), 1 -10 | MR 2984123

[8] L.V. Kantorovich, On the translocation of masses, J. Math. Sci. 133 no. 4 (2006), 1381 -1382 | MR 2117876 | Zbl 1080.49507

[9] R.J. Mccann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 no. 3 (2001), 589 -608 | MR 1844080 | Zbl 1011.58009

[10] G. Monge, Mémoire sur la théorie des déblais et des remblais, De l'Imprimerie Royale (1781)

[11] S.-I. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805 -828 | MR 2341840 | Zbl 1176.28016

[12] A.M. Srivastava, A Course on Borel Sets, Springer (1998) | MR 1619545 | Zbl 0903.28001

[13] T. Rajala, K.T. Sturm, Non-branching geodesics and optimal maps in strong $𝖢𝖣 (K, \infty)$ -spaces, Calc. Var. Partial Differ. Equ. 50 no. 3–4 (2014), 831 -846 | MR 3216835 | Zbl 1296.53088

[14] K.T. Sturm, On the geometry of metric measure spaces.II, Acta Math. 196 no. 1 (2006), 133 -177 | MR 2237207

[15] C. Villani, Optimal Transport, Old and New, Springer (2008) | MR 2459454 | Zbl 1158.53036