Mass transport problem and derivation

Belili, Nacereddine ; Heinich, Henri

Applicationes Mathematicae, Tome 26 (1999), p. 299-314 / Harvested from The Polish Digital Mathematics Library

Résumé

A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function f and the fact that the probability measure $λ_{f*}:=λ◦(f*)^{-1}$, where f*(x):=(x,f(x)) and λ is the Lebesgue measure, has the transport property.

Détail

Publié le : 1999-01-01

Zbl 0998.60012

EUDML-ID : urn:eudml:doc:219241

Bibliographie

[000] [1] T. Abdellaoui et H. Heinich, Sur la distance de deux lois dans le cas vectoriel, C. R. Acad. Sci. Paris 319 (1994), 397-400. | Zbl 0808.60008

[001] [2] N. Belili, A theorem of stochastic representation related to the Monge-Kantorovich problem, Statist. Probab. Lett. 41 (1999), 139-143. | Zbl 0916.60005

[002] [3] C. L. Belna, M. J. Cargo, M. J. Evans and P. D. Humke, Analogues of the Denjoy-Young-Saks theorem, Trans. Amer. Math. Soc. 271 (1982), 253-260. | Zbl 0486.26003

[003] [4] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris 305 (1987), 805-808. | Zbl 0652.26017

[004] [5] Y. Brenier, The dual least action problem for an ideal incompressible fluid, Arch. Rational Mech. Anal. 122 (1991), 323-351. | Zbl 0797.76006

[005] [6] Y. Brenier, Polar factorization and monotone rearragement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417.

[006] [7] A. Bruckner, Differentiation of Real Functions, CRM Monogr. Ser. 5, Amer. Math. Soc., 1994.

[007] [8] J. A. Cuesta-Albertos and C. Matrán-Bea, Notes on the Wasserstein metric in Hilbert spaces, Ann. Probab. 17 (1989), 1264-1276. | Zbl 0688.60011

[008] [9] J. A. Cuesta-Albertos, C. Matrán-Bea and A. Tuero-Diaz, On lower bounds for the $L^2$-Wasserstein metric in a Hilbert space, J. Theoret. Probab. 9 (1996), 263-283. | Zbl 0870.60005

[009] [10] J. A. Cuesta-Albertos, C. Matrán-Bea and A. Tuero-Diaz, Propreties of the optimal maps for the $l^2$-Monge-Kantorovich transportation problem, preprint, 1996. | Zbl 0870.60005

[010] [11] C. Dellacherie et P. A. Meyer, Probabilités et potentiel, Hermann, Paris, 1983.

[011] [12] A. Faure, Sur le théorème de Denjoy-Young-Saks, C. R. Acad. Sci. Paris 320 (1995), 415-418. | Zbl 0835.26006

[012] [13] W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), 113-161. | Zbl 0887.49017

[013] [14] K. M. Garg, Applications of Denjoy analogue. II: Local structure of level sets and Dini derivates. III: Distribution of various typical level sets, Acta Math. Acad. Sci. Hungar. 14 (1963), 183-195. | Zbl 0142.02104

[014] [15] H. Heinich et J. C. Lootgieter, Convergence des fonctions monotones, C. R. Acad. Sci. Paris 322 (1996), 869-874. | Zbl 0849.60027

[015] [16] L. V. Kantorovich, On the translocation of masses, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 199-201. | Zbl 0061.09705

[016] [17] L. V. Kantorovich, On a problem of Monge, Uspekhi Mat. Nauk 3 (1948), 225-226 (in Russian).

[017] [18] M. Knott and C. S. Smith, On the optimal transportation of distributions, J. Optim. Theory Appl. 52 (1987), 323-329. | Zbl 0586.49005

[018] [19] M. Knott and C. S. Smith, On Hoeffding-Fréchet bounds and cyclic monotone relations, J. Multivariate Anal. 40 (1992), 328-334. | Zbl 0745.62055

[019] [20] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), 309-323. | Zbl 0873.28009

[020] [21] R. J. McCann, personal communication, 1996.

[021] [22] G. Monge, Mémoire sur la théorie des déblais et des remblais, dans : Histoires de l'Académie Royale des Sciences de Paris, avec les mémoires de Mathématiques et de Physique pour la même année, 1781, 257-263.

[022] [23] S. T. Rachev, The Monge-Kantorovich mass transference problem and its stochastic applications, Theory Probab. Appl. 29 (1984), 647-676. | Zbl 0581.60010

[023] [24] S. T. Rachev and L. Rüschendorf, A characterization of random variables with minimum $l^2$-distance, J. Multivariate Anal. 32 (1990), 48-54. | Zbl 0688.62034

[024] [25] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, Springer, New York, 1998. | Zbl 0990.60500

[025] [26] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, 1970. | Zbl 0193.18401

[026] [27] V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Transl. 71 (1952), 1-54.

[027] [28] L. Rüschendorf, Fréchet bounds and their applications, in: S. Kotz, G. Dall'Aglio and G. Salinetti (eds.), Advances in Probability Distributions with Given Marginals: Beyond the Copulas, Kluwer, Dordrecht, 1991, 141-176.

[028] [29] L. Rüschendorf, Optimal solutions of multivariate coupling problems, Appl. Math. (Warsaw) 23 (1995), 325-338. | Zbl 0844.62047

[029] [30] S. Saks, Sur les nombres dérivés des fonctions, Fund. Math. 5 (1924), 98-104.

[030] [31] A. Tuero-Diaz, On the stochastic convergence of representations based on Wasserstein metrics, Ann. Probab. 21 (1993), 72-85. | Zbl 0770.60012

[031] [32] L. Zajiček, On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (1979), 340-348. | Zbl 0429.46007

[032] [33] L. Zajiček, On the symmetry of Dini derivates of arbitrary functions, Comment. Math. Univ. Carolin. 22 (1981), 195-209. | Zbl 0462.26003