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A generalized dual maximizer for the Monge–Kantorovich transport problem

  • Mathias Beiglböck (a1), Christian Léonard (a2) and Walter Schachermayer (a3)

Abstract

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.

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[1] J. Aaronson, An introduction to infinite ergodic theory, in Math. Surveys Monogr., Amer. Math. Soc. Providence, RI 50 (1997).
[2] Aaronson, J. and Keane, M., The visits to zero of some deterministic random walks. Proc. London Math. Soc. 44 (1982) 535553.
[3] Ambrosio, L. and Pratelli, A., Existence and stability results in the L1-theory of optimal transportation, CIME Course Lect. Notes Math. 1813 (2003) 123160.
[4] Beiglböck, M., Goldstern, M., Maresh, G. and Schachermayer, W., Optimal and better transport plans. J. Funct. Anal. 256 (2009) 19071927.
[5] M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge–Kantorovich transport problem. Submitted (2009).
[6] 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)
[7] Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991) 375417.
[8] Beiglböck, M. and Schachermayer, W., Duality for Borel measurable cost functions. Trans. Amer. Math. Soc. 363 (2011) 42034224.
[9] Probabilités, I (Univ. Rennes, Rennes, 1976). Exp. No. 5, Dépt. Math. Informat., Univ. Rennes, Rennes (1976) 7.
[10] Cafarelli, L. and McCann, R.J., Free boundaries in optimal transport and Monge–Ampere obstacle problems. Ann. of Math. 171 (2010) 673730.
[11] de Acosta, A., Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Probab. 10 (1982) 346373.
[12] Decreusefond, L., Wasserstein distance on configuration space. Potential Anal. 28 (2008) 283300.
[13] Decreusefond, L., Joulin, A. and Savy, N., Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stochastic Anal. 4 (2010) 377399.
[14] 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).
[15] R.M. Dudley, Real analysis and probability, Cambridge University Press, Cambridge. Cambridge Studies in Adv. Math. 74 (2002). Revised reprint of the 1989 original.
[16] 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) 610.
[17] Figalli, A., The optimal partial transport problem. Arch. Rational Mech. Anal. 195 (2010) 533560.
[18] Feyel, D. and Üstünel, A.S., Measure transport on Wiener space and the Girsanov theorem. C. R. Math. Acad. Sci. Paris 334 (2002) 10251028.
[19] Feyel, D. and Üstünel, A.S., Monge-Kantorovitch measure transportation and Monge–Ampère equation on Wiener space. Probab. Theory Relat. Fields 128 (2004) 347385.
[20] 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) 4974.
[21] 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) 2955.
[22] Gaffke, N. and Rüschendorf, L., On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optim. 12 (1981) 123135.
[23] Gangbo, W. and McCann, R.J., The geometry of optimal transportation. Acta Math. 177 (1996) 113161.
[24] Kantorovich, L.V., On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS 37 (1942) 199201.
[25] Kantorovič, L.V. and Rubinšteĭn, G.Š., On a space of completely additive functions. Vestnik Leningrad. Univ. 13 (1958) 5259.
[26] Kellerer, H., Duality theorems for marginal problems. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 67 (1984) 399432.
[27] Léonard, C., A saddle-point approach to the Monge–Kantorovich transport problem. ESAIM : COCV 17 (2011) 682704.
[28] McCann, R., Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (1995) 309323.
[29] Mikami, T., A simple proof of duality theorem for Monge–Kantorovich problem. Kodai Math. J. 29 (2006) 14.
[30] Mikami, T. and Thieullen, M., Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl. 116 (2006) 18151835.
[31] Ramachandran, D. and Rüschendorf, L., A general duality theorem for marginal problems. Probab. Theory Relat. Fields 101 (1995) 311319.
[32] Ramachandran, D. and Rüschendorf, L., Duality and perfect probability spaces. Proc. Amer. Math. Soc. 124 (1996) 22232228.
[33] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I : Functional Analysis. Academic Press (1980).
[34] Rüschendorf, L., On c-optimal random variables. Stat. Probab. Lett. 27 (1996) 267270.
[35] Schmidt, K., A cylinder flow arising from irregularity of distribution. Compositio Math. 36 (1978) 225232.
[36] Schachermayer, W. and Teichman, J., Characterization of optimal transport plans for the Monge–Kantorovich problem. Proc. Amer. Math. Soc. 137 (2009) 519529.
[37] Szulga, A., On minimal metrics in the space of random variables. Teor. Veroyatnost. i Primenen. 27 (1982) 401405.
[38] 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) 897900.
[39] 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) 207234.
[40] C. Villani, Topics in Optimal Transportation, in Graduate Studies in Mathematics. Amer. Math. Soc., Providence RI 58 (2003).
[41] C. Villani, Optimal Transport, Old and New, in Grundlehren der mathematischen Wissenschaften. Springer 338 (2009).

Keywords

A generalized dual maximizer for the Monge–Kantorovich transport problem

  • Mathias Beiglböck (a1), Christian Léonard (a2) and Walter Schachermayer (a3)

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