The dual attainment of the Monge-Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be polish and equipped with Borel probability measures μ and ν. The transport cost function c : X × Y → [0,∞] is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel's perturbation technique.
Keywords: optimal transport, duality in function spaces, Fenchel's perturbation technique
@article{PS_2012__16__306_0,
author = {Beiglb\"ock, Mathias and L\'eonard, Christian and Schachermayer, Walter},
title = {A generalized dual maximizer for the {Monge-Kantorovich} transport problem},
journal = {ESAIM: Probability and Statistics},
pages = {306--323},
year = {2012},
publisher = {EDP Sciences},
volume = {16},
doi = {10.1051/ps/2011163},
mrnumber = {2956577},
zbl = {1263.49057},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2011163/}
}
TY - JOUR AU - Beiglböck, Mathias AU - Léonard, Christian AU - Schachermayer, Walter TI - A generalized dual maximizer for the Monge-Kantorovich transport problem JO - ESAIM: Probability and Statistics PY - 2012 SP - 306 EP - 323 VL - 16 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2011163/ DO - 10.1051/ps/2011163 LA - en ID - PS_2012__16__306_0 ER -
%0 Journal Article %A Beiglböck, Mathias %A Léonard, Christian %A Schachermayer, Walter %T A generalized dual maximizer for the Monge-Kantorovich transport problem %J ESAIM: Probability and Statistics %D 2012 %P 306-323 %V 16 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps/2011163/ %R 10.1051/ps/2011163 %G en %F PS_2012__16__306_0
Beiglböck, Mathias; Léonard, Christian; Schachermayer, Walter. A generalized dual maximizer for the Monge-Kantorovich transport problem. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 306-323. doi: 10.1051/ps/2011163
[1] , An introduction to infinite ergodic theory, in Math. Surveys Monogr., Amer. Math. Soc. Providence, RI 50 (1997). | Zbl | MR
[2] and , The visits to zero of some deterministic random walks. Proc. London Math. Soc. 44 (1982) 535-553. | Zbl | MR
[3] and , Existence and stability results in the L1-theory of optimal transportation, CIME Course Lect. Notes Math. 1813 (2003) 123-160. | Zbl | MR
[4] , , and , Optimal and better transport plans. J. Funct. Anal. 256 (2009) 1907-1927. | Zbl | MR
[5] , and , A general duality theorem for the Monge-Kantorovich transport problem. Submitted (2009). | Zbl
[6] , and , On the duality of the Monge-Kantorovich transport problem, in Summer school on optimal transport. Séminaires et Congrès, Société Mathématique de France, Institut Fourier, Grenoble (2009)
[7] , Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991) 375-417. | Zbl | MR
[8] and , Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. 363 (2011) 4203-4224. | Zbl | MR
[9] (Univ. Rennes, Rennes, 1976). Exp. No. 5, Dépt. Math. Informat., Univ. Rennes, Rennes (1976) 7.
[10] and , Free boundaries in optimal transport and Monge-Ampere obstacle problems. Ann. of Math. 171 (2010) 673-730. | Zbl | MR
[11] , Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Probab. 10 (1982) 346-373. | Zbl | MR
[12] , Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283-300. | Zbl | MR
[13] , and , Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stochastic Anal. 4 (2010) 377-399. | MR
[14] , Probabilities and metrics, Convergence of laws on metric spaces, with a view to statistical testing, No. 45. Matematisk Institut, Aarhus Universitet, Aarhus. Lect. Notes Ser. (1976). | Zbl | MR
[15] , Real analysis and probability, Cambridge University Press, Cambridge. Cambridge Studies in Adv. Math. 74 (2002). Revised reprint of the 1989 original. | Zbl | MR
[16] , Sur le théorème de Kantorovich-Rubinstein dans les espaces polonais in Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French). Lect. Notes Math. 850 (1981) 6-10. | Zbl | MR | Numdam
[17] , The optimal partial transport problem. Arch. Rational Mech. Anal. 195 (2010) 533-560. | Zbl | MR
[18] and , Measure transport on Wiener space and the Girsanov theorem. C. R. Math. Acad. Sci. Paris 334 (2002) 1025-1028. | Zbl | MR
[19] and , Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields 128 (2004) 347-385. | Zbl | MR
[20] and , Monge-Kantorovitch measure transportation, Monge-Ampère equation and the Itô calculus, in Stochastic analysis and related topics in Kyoto. Adv. Stud. Pure Math. Math. Soc. Japan 41 (2004) 49-74. | Zbl
[21] and , Solution of the Monge-Ampère equation on Wiener space for general log-concave measures. J. Funct. Anal. 232 (2006) 29-55. | Zbl | MR
[22] and , On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optim. 12 (1981) 123-135. | Zbl | MR
[23] and , The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | Zbl | MR
[24] , On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS 37 (1942) 199-201. | Zbl | MR
[25] and , On a space of completely additive functions. Vestnik Leningrad. Univ. 13 (1958) 52-59. | Zbl
[26] , Duality theorems for marginal problems. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 67 (1984) 399-432. | Zbl | MR
[27] , A saddle-point approach to the Monge-Kantorovich transport problem. ESAIM : COCV 17 (2011) 682-704. | Zbl | MR | Numdam
[28] , Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (1995) 309-323. | Zbl | MR
[29] , A simple proof of duality theorem for Monge-Kantorovich problem. Kodai Math. J. 29 (2006) 1-4. | Zbl | MR
[30] and , Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116 (2006) 1815-1835. | Zbl | MR
[31] and , A general duality theorem for marginal problems. Probab. Theory Relat. Fields 101 (1995) 311-319. | Zbl | MR
[32] and , Duality and perfect probability spaces. Proc. Amer. Math. Soc. 124 (1996) 2223-2228. | Zbl | MR
[33] and , Methods of Modern Mathematical Physics, I : Functional Analysis. Academic Press (1980). | Zbl | MR
[34] , On c-optimal random variables. Stat. Probab. Lett. 27 (1996) 267-270. | Zbl | MR
[35] , A cylinder flow arising from irregularity of distribution. Compositio Math. 36 (1978) 225-232. | Zbl | MR | Numdam
[36] and , Characterization of optimal transport plans for the Monge-Kantorovich problem. Proc. Amer. Math. Soc. 137 (2009) 519-529. | Zbl | MR
[37] , On minimal metrics in the space of random variables. Teor. Veroyatnost. i Primenen. 27 (1982) 401-405. | Zbl | MR
[38] , A necessary, and sufficient condition for invertibility of adapted perturbations of identity on Wiener space. C. R. Acad. Sci. Paris, Ser. I 346 (2008) 897-900. | Zbl
[39] and , Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields 139 (2007) 207-234. | Zbl | MR
[40] , Topics in Optimal Transportation, in Graduate Studies in Mathematics. Amer. Math. Soc., Providence RI 58 (2003). | Zbl | MR
[41] , Optimal Transport, Old and New, in Grundlehren der mathematischen Wissenschaften. Springer 338 (2009). | Zbl | MR
Cité par Sources :
