A generalized dual maximizer for the Monge–Kantorovich transport problem∗

Mathias Beiglböck; Christian Léonard; Walter Schachermayer

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 306-323
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Beiglböck, Mathias, Léonard, Christian, and Schachermayer, Walter. "A generalized dual maximizer for the Monge–Kantorovich transport problem∗." ESAIM: Probability and Statistics 16 (2012): 306-323. <http://eudml.org/doc/222460>.

@article{Beiglböck2012,
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.},
author = {Beiglböck, Mathias, Léonard, Christian, Schachermayer, Walter},
journal = {ESAIM: Probability and Statistics},
keywords = {Optimal transport; duality in function spaces; Fenchel’s perturbation technique; optimal transport; Fenchel's perturbation technique},
language = {eng},
month = {7},
pages = {306-323},
publisher = {EDP Sciences},
title = {A generalized dual maximizer for the Monge–Kantorovich transport problem∗},
url = {http://eudml.org/doc/222460},
volume = {16},
year = {2012},
}

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
DA - 2012/7//
PB - EDP Sciences
VL - 16
SP - 306
EP - 323
AB - 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.
LA - eng
KW - Optimal transport; duality in function spaces; Fenchel’s perturbation technique; optimal transport; Fenchel's perturbation technique
UR - http://eudml.org/doc/222460
ER -

References

top
  1. J. Aaronson, An introduction to infinite ergodic theory, in Math. Surveys Monogr., Amer. Math. Soc. Providence, RI 50 (1997).  Zbl0882.28013
  2. J. Aaronson and M. Keane, The visits to zero of some deterministic random walks. Proc. London Math. Soc.44 (1982) 535–553.  Zbl0489.60006
  3. L. Ambrosio and A. Pratelli, Existence and stability results in the L1-theory of optimal transportation, CIME Course Lect. Notes Math. 1813 (2003) 123–160.  Zbl1065.49026
  4. M. Beiglböck, M. Goldstern, G. Maresh and W. Schachermayer, Optimal and better transport plans. J. Funct. Anal.256 (2009) 1907–1927.  Zbl1157.49019
  5. M. Beiglböck, C. Léonard and W. Schachermayer, A general duality theorem for the Monge–Kantorovich transport problem. Submitted (2009).  Zbl1270.49045
  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)  Zbl1333.49064
  7. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math.44 (1991) 375–417.  Zbl0738.46011
  8. M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. Trans. Amer. Math. Soc.363 (2011) 4203–4224.  Zbl1228.49046
  9. Probabilités, I (Univ. Rennes, Rennes, 1976). Exp. No. 5, Dépt. Math. Informat., Univ. Rennes, Rennes (1976) 7.  
  10. L. Cafarelli and R.J. McCann, Free boundaries in optimal transport and Monge–Ampere obstacle problems. Ann. of Math.171 (2010) 673–730.  Zbl1196.35231
  11. A. de Acosta, Invariance principles in probability for triangular arrays of B-valued random vectors and some applications. Ann. Probab.10 (1982) 346–373.  Zbl0499.60009
  12. L. Decreusefond, Wasserstein distance on configuration space. Potential Anal.28 (2008) 283–300.  Zbl1144.60004
  13. L. Decreusefond, A. Joulin and N. Savy, Upper bounds on Rubinstein distances on configuration spaces and applications. Commun. Stochastic Anal.4 (2010) 377–399.  Zbl1331.60083
  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).  Zbl0355.60004
  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.  Zbl1023.60001
  16. X. Fernique, 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.  
  17. A. Figalli, The optimal partial transport problem. Arch. Rational Mech. Anal.195 (2010) 533–560.  Zbl1245.49059
  18. D. Feyel and A.S. Üstünel, Measure transport on Wiener space and the Girsanov theorem. C. R. Math. Acad. Sci. Paris334 (2002) 1025–1028.  Zbl1036.60004
  19. D. Feyel and A.S. Üstünel, Monge-Kantorovitch measure transportation and Monge–Ampère equation on Wiener space. Probab. Theory Relat. Fields128 (2004) 347–385.  Zbl1055.60052
  20. D. Feyel and A.S. Üstünel, 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. Japan41 (2004) 49–74.  
  21. D. Feyel and A.S. Üstünel, Solution of the Monge-Ampère equation on Wiener space for general log-concave measures. J. Funct. Anal.232 (2006) 29–55.  Zbl1099.60042
  22. N. Gaffke and L. Rüschendorf, On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optim.12 (1981) 123–135.  Zbl0467.60004
  23. W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161.  Zbl0887.49017
  24. L.V. Kantorovich, On the translocation of masses. C. R. (Dokl.) Acad. Sci. URSS37 (1942) 199–201.  Zbl0061.09705
  25. L.V. Kantorovič and G.Š. Rubinšteĭn, On a space of completely additive functions. Vestnik Leningrad. Univ.13 (1958) 52–59.  
  26. H. Kellerer, Duality theorems for marginal problems. Z. Wahrscheinlichkeitstheorie Verw. Gebiete67 (1984) 399–432.  Zbl0535.60002
  27. C. Léonard, A saddle-point approach to the Monge–Kantorovich transport problem. ESAIM : COCV17 (2011) 682–704.  Zbl1234.46058
  28. R. McCann, Existence and uniqueness of monotone measure-preserving maps. Duke Math. J.80 (1995) 309–323.  Zbl0873.28009
  29. T. Mikami, A simple proof of duality theorem for Monge–Kantorovich problem. Kodai Math. J.29 (2006) 1–4.  Zbl1113.49039
  30. T. Mikami and M. Thieullen, Duality theorem for the stochastic optimal control problem. Stoch. Proc. Appl.116 (2006) 1815–1835.  Zbl1118.93056
  31. D. Ramachandran and L. Rüschendorf, A general duality theorem for marginal problems. Probab. Theory Relat. Fields101 (1995) 311–319.  Zbl0818.60001
  32. D. Ramachandran and L. Rüschendorf, Duality and perfect probability spaces. Proc. Amer. Math. Soc.124 (1996) 2223–2228.  Zbl0863.60005
  33. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I : Functional Analysis. Academic Press (1980).  Zbl0459.46001
  34. L. Rüschendorf, On c-optimal random variables. Stat. Probab. Lett.27 (1996) 267–270.  Zbl0847.62046
  35. K. Schmidt, A cylinder flow arising from irregularity of distribution. Compositio Math.36 (1978) 225–232.  Zbl0388.28019
  36. W. Schachermayer and J. Teichman, Characterization of optimal transport plans for the Monge–Kantorovich problem. Proc. Amer. Math. Soc.137 (2009) 519–529.  Zbl1165.49015
  37. A. Szulga, On minimal metrics in the space of random variables. Teor. Veroyatnost. i Primenen.27 (1982) 401–405.  Zbl0493.60016
  38. A.S. Üstünel, A necessary, and sufficient condition for invertibility of adapted perturbations of identity on Wiener space. C. R. Acad. Sci. Paris, Ser. I346 (2008) 897–900.  Zbl1148.60033
  39. A.S. Üstünel and M. Zakai, Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields139 (2007) 207–234.  Zbl1120.60057
  40. C. Villani, Topics in Optimal Transportation, in Graduate Studies in Mathematics. Amer. Math. Soc., Providence RI 58 (2003).  Zbl1106.90001
  41. C. Villani, Optimal Transport, Old and New, in Grundlehren der mathematischen Wissenschaften. Springer 338 (2009).  

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