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A generalized dual maximizer for the Monge–Kantorovichtransport problem∗

Published online by Cambridge University Press:  16 July 2012

Mathias Beiglböck ,
Christian Léonard  and
Walter Schachermayer
Mathias Beiglböck
Affiliation:
University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria. [email protected]
Christian Léonard
Affiliation:
Modal-X, Université Paris Ouest, Bât. G, 200 av. de la République, 92001 Nanterre, France; [email protected]
Walter Schachermayer
Affiliation:
University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Vienna, Austria; [email protected]
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Abstract

The dual attainment of the Monge–Kantorovich transport problem is analyzed in a generalsetting. The spaces X,Y are assumed to be polish and equipped with Borelprobability measures μ and ν. The transport costfunction c : X × Y →  [0,∞]  is assumedto be Borel measurable. We show that a dual optimizer always exists, provided we interpretit as a projective limit of certain finitely additive measures. Our methods are functionalanalytic and rely on Fenchel’s perturbation technique.

Keywords

Optimal transportduality in function spacesFenchel’s perturbation technique
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 306 - 323
Copyright
© EDP Sciences, SMAI, 2012

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References

J. Aaronson, An introduction to infinite ergodic theory, in Math. Surveys Monogr., Amer. Math. Soc. Providence, RI 50 (1997).
Aaronson, J. and Keane, M., The visits to zero of some deterministic random walks. Proc. London Math. Soc. 44 (1982) 535–553. Google Scholar
Ambrosio, L. and Pratelli, A., Existence and stability results in the L1-theory of optimal transportation, CIME Course Lect. Notes Math. 1813 (2003) 123–160. CrossRefGoogle Scholar
Beiglböck, M., Goldstern, M., Maresh, G. and Schachermayer, W., Optimal and better transport plans. J. Funct. Anal. 256 (2009) 1907–1927. Google Scholar
M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge–Kantorovich transport problem. Submitted (2009).
M. Beiglböck, C. Léonard and W. Schachermayer, 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)
Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991) 375–417. Google Scholar
Beiglböck, M. and Schachermayer, W., Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. 363 (2011) 4203–4224. Google Scholar
Probabilités, I (Univ. Rennes, Rennes, 1976). Exp. No. 5, Dépt. Math. Informat., Univ. Rennes, Rennes (1976) 7.
Cafarelli, L. and McCann, R.J., Free boundaries in optimal transport and Monge–Ampere obstacle problems. Ann. of Math. 171 (2010) 673–730. Google Scholar
de Acosta, A., Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Probab. 10 (1982) 346–373. Google Scholar
Decreusefond, L., Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283–300. Google Scholar
Decreusefond, L., Joulin, A. and Savy, N., Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stochastic Anal. 4 (2010) 377–399. Google Scholar
R.M. Dudley, 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).
R.M. Dudley, Real analysis and probability, Cambridge University Press, Cambridge. Cambridge Studies in Adv. Math. 74 (2002). Revised reprint of the 1989 original.
Fernique, X., 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. Google Scholar
Figalli, A., The optimal partial transport problem. Arch. Rational Mech. Anal. 195 (2010) 533–560. Google Scholar
Feyel, D. and Üstünel, A.S., Measure transport on Wiener space and the Girsanov theorem. C. R. Math. Acad. Sci. Paris 334 (2002) 1025–1028. Google Scholar
Feyel, D. and Üstünel, A.S., Monge-Kantorovitch measure transportation and Monge–Ampère equation on Wiener space. Probab. Theory Relat. Fields 128 (2004) 347–385. Google Scholar
Feyel, D. and Üstünel, A.S., 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. Google Scholar
Feyel, D. and Üstünel, A.S., Solution of the Monge-Ampère equation on Wiener space for general log-concave measures. J. Funct. Anal. 232 (2006) 29–55. Google Scholar
Gaffke, N. and Rüschendorf, L., On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optim. 12 (1981) 123–135. Google Scholar
Gangbo, W. and McCann, R.J., The geometry of optimal transportation. Acta Math. 177 (1996) 113–161. Google Scholar
Kantorovich, L.V., On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS 37 (1942) 199–201. Google Scholar
Kantorovič, L.V. and Rubinšteĭn, G.Š., On a space of completely additive functions. Vestnik Leningrad. Univ. 13 (1958) 52–59. Google Scholar
Kellerer, H., Duality theorems for marginal problems. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 67 (1984) 399–432. Google Scholar
Léonard, C., A saddle-point approach to the Monge–Kantorovich transport problem. ESAIM : COCV 17 (2011) 682–704. Google Scholar
McCann, R., Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (1995) 309–323. Google Scholar
Mikami, T., A simple proof of duality theorem for Monge–Kantorovich problem. Kodai Math. J. 29 (2006) 1–4. Google Scholar
Mikami, T. and Thieullen, M., Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116 (2006) 1815–1835. Google Scholar
Ramachandran, D. and Rüschendorf, L., A general duality theorem for marginal problems. Probab. Theory Relat. Fields 101 (1995) 311–319. Google Scholar
Ramachandran, D. and Rüschendorf, L., Duality and perfect probability spaces. Proc. Amer. Math. Soc. 124 (1996) 2223–2228. Google Scholar
M. Reed and B. Simon, Methods of Modern Mathematical Physics, I : Functional Analysis. Academic Press (1980).
Rüschendorf, L., On c-optimal random variables. Stat. Probab. Lett. 27 (1996) 267–270. Google Scholar
Schmidt, K., A cylinder flow arising from irregularity of distribution. Compositio Math. 36 (1978) 225–232. Google Scholar
Schachermayer, W. and Teichman, J., Characterization of optimal transport plans for the Monge–Kantorovich problem. Proc. Amer. Math. Soc. 137 (2009) 519–529. Google Scholar
Szulga, A., On minimal metrics in the space of random variables. Teor. Veroyatnost. i Primenen. 27 (1982) 401–405. Google Scholar
Üstünel, A.S., 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. Google Scholar
Üstünel, A.S. and Zakai, M., Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields 139 (2007) 207–234. Google Scholar
C. Villani, Topics in Optimal Transportation, in Graduate Studies in Mathematics. Amer. Math. Soc., Providence RI 58 (2003).
C. Villani, Optimal Transport, Old and New, in Grundlehren der mathematischen Wissenschaften. Springer 338 (2009).