Let X be a Polish space, \( \mathcal{P} \)(X) be the set of Borel probability measures on X, and T : X → X be a homeomorphism. We prove that for the simplex Dom ⊆ \( \mathcal{P} \)(X) of all T -invariant measures, the Kantorovich metric on 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 an the class of ergodic decomposable simplices.
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References
C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York (1985).
L. Ambrosio and N. Gigli, “A user’s guide to optimal transport,” Lect. Notes Math., 2062, 1–155 (2013).
M. Beiglböck and C. Griessler, “An optimality principle with applications in optimal transport,” arXiv:1404.7054 (2014).
V. Bogachev, Measure Theory, Vol. 2, Springer, New York (2007).
V. I. Bogachev and A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives,” Russian Math. Surveys, 67, No. 5(785), 3–110 (2012).
O. Chodosh, “Optimal transport and Ricci curvature: Wasserstein space over the interval,” arXiv:1105.2883 (2011).
M. Colombo and S. Di Marino, “Equality between Monge and Kantorovich multimarginal problems with Coulomb cost,” Ann. Mat. Pura Appl., 194, No. 2, 307–320 (2015).
M. I. Cortez and J. Rivera-Letelier, “Choquet simplices as spaces of invariant probability measures on post-critical sets,” Ann. Inst. H. Poincaré Anal. Non Linéare, 27, No. 1, 95–115 (2010).
T. Downarowicz, “The Choquet simplex of invariant measures for minimal flows,” Israel J. Math., 74, No. 2–3, 241–256 (1991).
E. B. Dynkin, “Sufficient statistics and extreme points,” Ann. Probab., 6, No. 5, 705–730 (1978).
M. Einsiedler and T. Ward, Ergodic Theory: With a View Towards Number Theory, Springer-Verlag (2011).
M. Gaudard and D. Hadwin, “Sigma-algebras on spaces of probability measures,” Scand. J. Statist., 16, No. 2, 169–175 (1989).
C. Himmelberg, “Measurable relations,” Fund. Math., 87, No. 1, 53–72 (1975).
J. Kerstan and A. Wakolbinger, “Ergodic decomposition of probability laws,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 56, No. 3, 339—414 (1981).
A. V. Kolesnikov and D. Zaev, “Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry,” arXiv:1303.7255 (2013).
A. Moameni, “Invariance properties of the Monge–Kantorovich mass transport problem,” arXiv:1311.7051 (2013).
K. R. Parthasarathy, Probability Measures on Metric Spaces, AMS Chelsea Publishing (1967).
R. Phelps, Lectures on Choquet’s Theorem, Springer-Verlag, Berlin–Heidelberg (2001).
U. Rieder, “Measurable selection theorems for optimization problems,” Manuscripta Math., 24, No. 1, 1151–131 (1978).
A. M. Vershik, “Kantorovich metric: initial history and little-known applications,” J. Math. Sci., 133, No. 4, 1410–1417 (2006).
A. M. Vershik, P. B. Zatitskiy, and F. V. Petrov, “Virtual continuity of measurable functions and its applications,” Russian Math. Surveys, 69, No. 6(420), 81–114 (2014).
A. M. Vershik, “Equipped graded graphs, projective limits of simplices, and their boundaries,” J. Math. Sci., 209, No. 6, 860–873 (2015).
C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin (2009).
D. Zaev, On the Monge–Kantorovich problem with additional linear constraints,” Mat. Zametki, 98, No. 5, 664–683 (2015).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 437, 2015, pp. 100–130.
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Zaev, D.A. On Ergodic Decompositions Related to the Kantorovich Problem. J Math Sci 216, 65–83 (2016). https://doi.org/10.1007/s10958-016-2888-9
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DOI: https://doi.org/10.1007/s10958-016-2888-9