Existence and uniqueness of optimal transport maps

Cavalletti, Fabio; Huesmann, Martin

doi:10.1016/j.anihpc.2014.09.006

Cavalletti, Fabio ; Huesmann, Martin

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 6, pp. 1367-1377.

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.

MR Zbl | 3 citations dans Numdam

DOI : 10.1016/j.anihpc.2014.09.006

Mots-clés : Optimal transport, Existence of maps, Uniqueness of maps, Measure contraction property

@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},
     pages = {1367--1377},
     publisher = {Elsevier},
     volume = {32},
     number = {6},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.09.006},
     mrnumber = {3425266},
     zbl = {1331.49063},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/}
}

TY  - JOUR
AU  - Cavalletti, Fabio
AU  - Huesmann, Martin
TI  - Existence and uniqueness of optimal transport maps
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 1367
EP  - 1377
VL  - 32
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/
DO  - 10.1016/j.anihpc.2014.09.006
LA  - en
ID  - AIHPC_2015__32_6_1367_0
ER  -

%0 Journal Article
%A Cavalletti, Fabio
%A Huesmann, Martin
%T Existence and uniqueness of optimal transport maps
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 1367-1377
%V 32
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/
%R 10.1016/j.anihpc.2014.09.006
%G en
%F 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) no. 6, pp. 1367-1377. doi : 10.1016/j.anihpc.2014.09.006. http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/

Bibliographie
Cité par

[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

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

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

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

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

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

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

Cité par Sources :