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 Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 2, Pages 403–409 (Mi tvp474)

Short Communications

On the Monge–Kantorovich duality theorem

D. Ramachandrana, L. Rüschendorfb

a California State University, Department of Mathematics and Statistics
b Institut füur Mathematische Stochastik, Albert-Ludwigs-Universität, Germany

Abstract: The Monge–Kantorovitch duality theorem has a variety of applications in probability theory, statistics, and mathematical economics. There has been extensive work to establish the duality theorem under general conditions. In this paper, by imposing a natural stability requirement on the Monge–Kantorovitch functional, we characterize the probability spaces (called strong duality spaces) which ensure the validity of the duality theorem. We prove that strong duality is equivalent to each one of (i) extension property, (ii) projection property, (iii) the charge extension property, and (iv) perfectness. The resulting characterization enables us to derive many useful properties that such spaces inherit from being perfect.

Keywords: duality theorem, marginals, perfect measure, charge extension, Marczewski function.

DOI: https://doi.org/10.4213/tvp474

Full text: PDF file (529 kB)

English version:
Theory of Probability and its Applications, 2001, 45:2, 350–356

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Citation: D. Ramachandran, L. Rüschendorf, “On the Monge–Kantorovich duality theorem”, Teor. Veroyatnost. i Primenen., 45:2 (2000), 403–409; Theory Probab. Appl., 45:2 (2001), 350–356

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp474
• https://doi.org/10.4213/tvp474
• http://mi.mathnet.ru/eng/tvp/v45/i2/p403

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Blomker D., Hairer M., “Multiscale expansion of invariant measures for SPDEs”, Communications in Mathematical Physics, 251:3 (2004), 515–555
2. Gonzalez-Hernandez J., Gabriel J.R., “On the consistency of the mass transfer problem”, Operations Research Letters, 34:4 (2006), 382–386
3. V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890
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