Duality for Borel measurable cost functions

Beiglböck, Mathias; Schachermayer, Walter

doi:10.1090/S0002-9947-2011-05174-3

Duality for Borel measurable cost functions
HTML articles powered by AMS MathViewer

by Mathias Beiglböck and Walter Schachermayer PDF

Trans. Amer. Math. Soc. 363 (2011), 4203-4224 Request permission

Abstract:

We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if $c:X\times Y\to [0,\infty )$ is an arbitrary Borel measurable cost function on the product of Polish spaces $X,Y$. In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a “subsidy” function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.

References

Luigi Ambrosio and Aldo Pratelli, Existence and stability results in the $L^1$ theory of optimal transportation, Optimal transportation and applications (Martina Franca, 2001) Lecture Notes in Math., vol. 1813, Springer, Berlin, 2003, pp. 123–160. MR 2006307, DOI 10.1007/978-3-540-44857-0_{5}
M. Beiglböck, M. Goldstern, G. Maresch, and W. Schachermayer. Optimal and better transport. arXiv:0802.0646v1, 2008.
J. M. Borwein and A. S. Lewis, Decomposition of multivariate functions, Canad. J. Math. 44 (1992), no. 3, 463–482. MR 1176365, DOI 10.4153/CJM-1992-030-9
Freddy Delbaen and Walter Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), no. 3, 463–520. MR 1304434, DOI 10.1007/BF01450498
Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), no. 2, 113–161. MR 1440931, DOI 10.1007/BF02392620
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Hans G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 4, 399–432. MR 761565, DOI 10.1007/BF00532047
Hans G. Kellerer, Duality theorems and probability metrics, Proceedings of the seventh conference on probability theory (Braşov, 1982) VNU Sci. Press, Utrecht, 1985, pp. 211–220. MR 867434
J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229. MR 210177, DOI 10.1007/BF02020976
A. Pratelli, On the sufficiency of $c$-cyclical monotonicity for optimality of transport plans, Math. Z. 258 (2008), no. 3, 677–690. MR 2369050, DOI 10.1007/s00209-007-0191-7
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497–510. MR 193549
D. Ramachandran and L. Rüschendorf, A general duality theorem for marginal problems, Probab. Theory Related Fields 101 (1995), no. 3, 311–319. MR 1324088, DOI 10.1007/BF01200499
D. Ramachandran and L. Rüschendorf, Duality and perfect probability spaces, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2223–2228. MR 1342043, DOI 10.1090/S0002-9939-96-03462-4
Svetlozar T. Rachev and Ludger Rüschendorf, Mass transportation problems. Vol. I, Probability and its Applications (New York), Springer-Verlag, New York, 1998. Theory. MR 1619170
Ludger Rüschendorf, On $c$-optimal random variables, Statist. Probab. Lett. 27 (1996), no. 3, 267–270. MR 1395577, DOI 10.1016/0167-7152(95)00078-X
Walter Schachermayer and Josef Teichmann, Characterization of optimal transport plans for the Monge-Kantorovich problem, Proc. Amer. Math. Soc. 137 (2009), no. 2, 519–529. MR 2448572, DOI 10.1090/S0002-9939-08-09419-7
H. Steinhaus. Sur les distances des points des ensembles de mesure positive. Fund. Math., 1:93–104, 1920.
Karl Stromberg, An elementary proof of Steinhaus’s theorem, Proc. Amer. Math. Soc. 36 (1972), 308. MR 308368, DOI 10.1090/S0002-9939-1972-0308368-4
Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483, DOI 10.1090/gsm/058
Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454, DOI 10.1007/978-3-540-71050-9