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 Zap. Nauchn. Sem. POMI, 2015, Volume 437, Pages 100–130 (Mi znsl6175)

On ergodic decompositions related to the Kantorovich problem

D. A. Zaev

Department of Mathematics, National Research University "Higher School of Economics", Moscow, Russia

Abstract: Let $X$ be a Polish space, $\mathcal P(X)$ be the set of Borel probability measures on $X$, and $T\colon X\to X$ be a homeomorphism. We prove that for the simplex $\mathrm{Dom}\subseteq\mathcal P(X)$ of all $T$-invariant measures, the Kantorovich metric on $\mathrm{Dom}$ can be reconstructed from its values on the set of extreme points. This fact is closely related to the following result: the invariant optimal transportation plan is a mixture of invariant optimal transportation plans between extreme points of the simplex. The latter result can be generalized to the case of the Kantorovich problem with additional linear constraints and the class of ergodic decomposable simplices.

Key words and phrases: Kantorovich problem, ergodic decomposition, Markov kernel.

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English version:
Journal of Mathematical Sciences (New York), 2016, 216:1, 65–83

Bibliographic databases:

UDC: 517.972

Citation: D. A. Zaev, “On ergodic decompositions related to the Kantorovich problem”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Zap. Nauchn. Sem. POMI, 437, POMI, St. Petersburg, 2015, 100–130; J. Math. Sci. (N. Y.), 216:1 (2016), 65–83

Citation in format AMSBIB
\Bibitem{Zae15} \by D.~A.~Zaev \paper On ergodic decompositions related to the Kantorovich problem \inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XXVI. Representation theory, dynamical systems, combinatorial methods \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 437 \pages 100--130 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6175} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3499910} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 216 \issue 1 \pages 65--83 \crossref{https://doi.org/10.1007/s10958-016-2888-9} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84969791827} 

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This publication is cited in the following articles:
1. Alexander V. Kolesnikov, Danila A. Zaev, “Exchangeable optimal transportation and log-concavity”, Theory Stoch. Process., 20(36):2 (2015), 54–62
2. A. V. Kolesnikov, D. A. Zaev, “Optimal transportation of processes with infinite Kantorovich distance: independence and symmetry”, Kyoto J. Math., 57:2 (2017), 293–324
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