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Cyclical monotonicity and the ergodic theorem

Published online by Cambridge University Press:  09 January 2014

MATHIAS BEIGLBÖCK
MATHIAS BEIGLBÖCK*
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria email [email protected]
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Abstract

It is well known that optimal transport plans are cyclically monotone. The reverse implication that cyclically monotone transport plans are optimal needs some assumptions and the proof is non-trivial even if the costs are given by the squared euclidean distance on ${ \mathbb{R} }^{n} $. We establish this result as a corollary to the ergodic theorem.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 710 - 713
Copyright
© Cambridge University Press, 2014 

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References

Ambrosio, L. and Pratelli, A.. Existence and stability results in the ${L}^{1} $ theory of optimal transportation. Optimal Transportation and Applications (Martina Franca, 2001) (Lecture Notes in Mathematics, 1813). Springer, Berlin, 2003, pp. 123–160.Google Scholar
Bianchini, S. and Caravenna, L.. On optimality of $c$-cyclically monotone transference plans. C. R. Math. Acad. Sci. Paris 348 (11–12) (2010), 613–618.Google Scholar
Beiglböck, M., Goldstern, M., Maresch, G. and Schachermayer, W.. Optimal and better transport plans. J. Funct. Anal. 256 (6) (2009), 1907–1927.Google Scholar
Gangbo, W. and McCann, R.. The geometry of optimal transportation. Acta Math. 177 (2) (1996), 113–161.Google Scholar
Kallenberg, O.. Foundations of Modern Probability (Probability and its Applications), 2nd edn. Springer, New York, 2002.CrossRefGoogle Scholar
Knott, M. and Smith, C.. On Hoeffding–Fréchet bounds and cyclic monotone relations. J. Multivariate Anal. 40 (2) (1992), 328–334.Google Scholar
Pratelli, A.. On the sufficiency of $c$-cyclical monotonicity for optimality of transport plans. Math. Z. 258 (3) (2008), 677–690.Google Scholar
Rockafellar, R.. Characterization of the subdifferentials of convex functions. Pacific J. Math. 17 (1966), 497–510.CrossRefGoogle Scholar
Rüschendorf, L.. On $c$-optimal random variables. Statist. Probab. Lett. 27 (3) (1996), 267–270.Google Scholar
Schachermayer, W. and Teichmann, J.. Characterization of optimal transport plans for the Monge–Kantorovich problem. Proc. Amer. Math. Soc. 137 (2) (2009), 519–529.CrossRefGoogle Scholar
Villani, C.. Topics in Optimal Transportation (Graduate Studies in Mathematics, 58). American Mathematical Society, Providence, RI, 2003.Google Scholar
Villani, C.. Optimal Transport: Old and New (Grundlehren der mathematischen Wissenschaften, 338). Springer, Berlin, 2009.Google Scholar