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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 5(407), Pages 3–110 (Mi umn9490)  

This article is cited in 76 scientific papers (total in 76 papers)

The Monge–Kantorovich problem: achievements, connections, and perspectives

V. I. Bogachevab, A. V. Kolesnikovc

a M. V. Lomonosov Moscow State University
b St. Tikhon's Orthodox University
c Higher School of Economics

Abstract: This article gives a survey of recent research related to the Monge–Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse applications of these problems in non-linear analysis, probability theory, and differential geometry are discussed.
Bibliography: 196 titles.

Keywords: Monge problem, Kantorovich problem, optimal transportation, transport inequality, Kantorovich–Rubinshtein metric.


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English version:
Russian Mathematical Surveys, 2012, 67:5, 785–890

Bibliographic databases:

UDC: 519.2+517.9
MSC: 28C20, 35J96, 49Q20, 60B05
Received: 20.06.2012

Citation: V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Uspekhi Mat. Nauk, 67:5(407) (2012), 3–110; Russian Math. Surveys, 67:5 (2012), 785–890

Citation in format AMSBIB
\by V.~I.~Bogachev, A.~V.~Kolesnikov
\paper The Monge--Kantorovich problem: achievements, connections, and perspectives
\jour Uspekhi Mat. Nauk
\yr 2012
\vol 67
\issue 5(407)
\pages 3--110
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 5
\pages 785--890

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    1. Cheridito P., Kiiski M., Proemel D.J., Soner H.M., “Martingale Optimal Transport Duality”, Math. Ann.  crossref  isi
    2. A. M. Vershik, “Long history of the Monge-Kantorovich transportation problem”, Math. Intelligencer, 35:4 (2013), 1–9  crossref  mathscinet  zmath  isi  scopus
    3. A. V. Kolesnikov, O. V. Kudryavtseva, T. Nagapetyan, “Remarks on Afriat's theorem and the Monge-Kantorovich problem”, J. Math. Econom., 49:6 (2013), 501–505  crossref  mathscinet  zmath  isi  elib  scopus
    4. J. Manfred, M. Lippi, A. Passerini, P. Frasconi, “Type Extension Trees for feature construction and learning in relational domains”, Artificial Intelligence, 204 (2013), 30–55  crossref  mathscinet  zmath  isi  scopus
    5. G. A. Vasiliev, V. M. Khametov, E. A. Shelemekh, “Conditions for the Discreteness of Extremal Probability Measures (the Finite-Dimensional Case)”, Math. Notes, 94:6 (2013), 963–967  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. A. M. Vershik, “Two ways to define compatible metrics on the simplex of measures”, J. Math. Sci. (N. Y.), 196:2 (2014), 138–143  mathnet  crossref  mathscinet
    7. A. V. Kolesnikov, S. Yu. Tikhonov, “Regularity of the Monge–Ampère equation in Besov's spaces”, Calc. Var., 49:3-4 (2014), 1187–1197  crossref  mathscinet  zmath  isi  elib  scopus
    8. A. V. Kolesnikov, M. Röckner, “On continuity equations in infinite dimensions with non-Gaussian reference measure”, J. Funct. Anal., 266:7 (2014), 4490–4537  crossref  mathscinet  zmath  isi  elib  scopus
    9. O. Butkovsky, “Subgeometric rates of convergence of Markov processes in the Wasserstein metric”, Ann. Appl. Probab., 24:2 (2014), 526–552  crossref  mathscinet  zmath  isi  elib  scopus
    10. S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro, “Scaling hypothesis for the Euclidean bipartite matching problem”, Phys. Rev. E, 90:1 (2014), 012118  crossref  isi  elib  scopus
    11. D. B. Bukin, “On the Monge and Kantorovich problems for distributions of diffusion processes”, Math. Notes, 96:5-6 (2014), 864–870  crossref  mathscinet  zmath  isi  elib  scopus
    12. V. I. Bogachev, A. I. Kirillov, S. V. Shaposhnikov, “The Kantorovich and variation distances between invariant measures of diffusions and nonlinear stationary Fokker-Planck-Kolmogorov equations”, Math. Notes, 96:5-6 (2014), 855–863  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    13. F. Cacciafesta, A.-S. de Suzzoni, “Continuity of the flow of KdV with regard to the Wasserstein metrics and application to an invariant measure”, J. Differential Equations, 259:3 (2015), 1024–1067  crossref  mathscinet  zmath  isi  elib  scopus
    14. V. I. Bogachev, A. V. Shaposhnikov, “Lower bounds for the Kantorovich distance”, Dokl. Math., 91:1 (2015), 91–93  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    15. O. A. Manita, M. S. Romanov, S. V. Shaposhnikov, “Uniqueness of probability solutions to nonlinear Fokker-Planck-Kolmogorov equation”, Dokl. Math., 91:2 (2015), 142–146  crossref  crossref  mathscinet  zmath  isi  elib  elib  scopus
    16. D. Zaev, “On the Monge–Kantorovich Problem with Additional Linear Constraints”, Math. Notes, 98:5 (2015), 725–741  mathnet  crossref  crossref  mathscinet  isi  elib
    17. D. A. Zaev, “On ergodic decompositions related to the Kantorovich problem”, J. Math. Sci. (N. Y.), 216:1 (2016), 65–83  mathnet  crossref  mathscinet
    18. O. A. Manita, M. S. Romanov, S. V. Shaposhnikov, “On uniqueness of solutions to nonlinear Fokker-Planck-Kolmogorov equations”, Nonlinear Anal., 128 (2015), 199–226  crossref  mathscinet  zmath  isi  scopus
    19. V. I. Bogachev, A. N. Kalinin, “A continuous cost function for which the minima in the Monge and Kantorovich problems are not equal”, Dokl. Math., 92:1 (2015), 452–455  crossref  mathscinet  zmath  isi  scopus
    20. V. I. Bogachev, F.-Y. Wang, A. V. Shaposhnikov, “Estimates of the Kantorovich Norm on Manifolds”, Dokl. Math., 92:1 (2015), 494–499  crossref  mathscinet  zmath  isi  scopus
    21. Wu Zong-min, Tian Zheng, “Distribution function estimates by Wasserstein metric and Bernstein approximation for $C^{-1}$ functions”, Appl. Math. J. Chinese Univ. Ser. B, 30:2 (2015), 141–150  crossref  mathscinet  isi  scopus
    22. Alexander V. Kolesnikov, Danila A. Zaev, “Exchangeable optimal transportation and log-concavity”, Theory Stoch. Process., 20(36):2 (2015), 54–62  mathnet  mathscinet  zmath
    23. V. I. Bogachev, “Distributions of polynomials on multidimensional and infinite-dimensional spaces with measures”, Russian Math. Surveys, 71:4 (2016), 703–749  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    24. D. B. Bukin, “On the Kantorovich Problem for Nonlinear Images of the Wiener Measure”, Math. Notes, 100:5 (2016), 660–665  mathnet  crossref  crossref  mathscinet  isi  elib
    25. V. I. Bogachev, A. F. Miftakhov, “On weak convergence of finite-dimensional and infinite-dimensional distributions of random processes”, Theory Stoch. Process., 21(37):1 (2016), 1–11  mathnet  mathscinet  zmath  elib
    26. S. G. Bobkov, “Proximity of probability distributions in terms of Fourier–Stieltjes transforms”, Russian Math. Surveys, 71:6 (2016), 1021–1079  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    32. I. I. Malofeev, “Measurable dependence of conditional measures on a parameter”, Dokl. Math., 94:2 (2016), 493–497  crossref  mathscinet  zmath  isi  elib  scopus
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    35. G. V. Riabov, “A representation for the Kantorovich–Rubinstein distance defined by the Cameron–Martin norm of a Gaussian measure on a Banach space”, Theory Stoch. Process., 21(37):2 (2016), 84–90  mathnet  mathscinet  zmath
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    40. J. M. Nichols, J. A. Spendelow, J. D. Nichols, “Using optimal transport theory to estimate transition probabilities in metapopulation dynamics”, Ecol. Model., 359 (2017), 311–319  crossref  isi
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    43. O. Butkovsky, M. Scheutzow, “Invariant measures for stochastic functional differential equations”, Electron. J. Probab., 22 (2017), 98, 23 pp.  crossref  mathscinet  zmath  isi
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    58. Carlini E., Silva F.J., “On the Discretization of Some Nonlinear Fokker-Planck-Kolmogorov Equations and Applications”, SIAM J. Numer. Anal., 56:4 (2018), 2148–2177  crossref  mathscinet  zmath  isi  scopus
    59. V. V. Kozlov, “The Monge problem of “piles and holes” on the torus and the problem of small denominators”, Siberian Math. J., 59:6 (2018), 1090–1093  mathnet  crossref  crossref  mathscinet  isi  elib
    60. Bogachev V.I. Shaposhnikov A.V. Shaposhnikov S.V., “Estimates For Solutions to Fokker-Planck-Kolmogorov Equations With Integrable Drifts”, Dokl. Math., 98:3 (2018), 559–563  crossref  mathscinet  zmath  isi  scopus
    61. Kazanskiy N.L., Kharitonov S.I., Kozlova I.N., Moiseev M.A., “The Connection Between the Phase Problem in Optics, Focusing of Radiation, and the Monge-Kantorovich Problem”, Comput. Opt., 42:4 (2018), 574–587  crossref  isi
    62. Dmitry V. Bakin, Elena P. Krugova, “Transportation costs for optimal and triangular transformations of Gaussian measures”, Theory Stoch. Process., 23(39):2 (2018), 21–32  mathnet
    63. Zhao P., Zhou Zh.-H., “Label Distribution Learning By Optimal Transport”, Thirty-Second Aaai Conference on Artificial Intelligence / Thirtieth Innovative Applications of Artificial Intelligence Conference / Eighth Aaai Symposium on Educational Advances in Artificial Intelligence, Assoc Advancement Artificial Intelligence, 2018, 4506–4513  isi
    64. Bronevich A.G., Rozenberg I.N., “the Kantorovich Problem and Wasserstein Metric in the Theory of Belief Functions”, Belief Functions: Theory and Applications, Belief 2018, Lecture Notes in Artificial Intelligence, 11069, eds. Destercke S., Denoeux T., Cuzzolin F., Martin A., Springer International Publishing Ag, 2018, 31–38  crossref  isi
    65. Bogachev V.I. Miftakhov A.F. Shaposhnikov S.V., “Differential Properties of Semigroups and Estimates of Distances Between Stationary Distributions of Diffusions”, Dokl. Math., 99:2 (2019), 175–180  crossref  isi
    66. Cioletti L., Silva E., Stadlbauer M., “Thermodynamic Formalism For Topological Markov Chains on Standard Borel Spaces”, Discret. Contin. Dyn. Syst., 39:11 (2019), 6277–6298  crossref  isi
    67. Gladkov N.A., Kolesnikov V A., Zimin A.P., “on Multistochastic Monge-Kantorovich Problem, Bitwise Operations, and Fractals”, Calc. Var. Partial Differ. Equ., 58:5 (2019), 173  crossref  isi
    68. Bogachev V.I., Malofeev I.I., “on the Kantorovich Problem With a Parameter”, Dokl. Math., 100:1 (2019), 349–353  crossref  isi
    69. Cobzas S., Miculescu R., Nicolae A., “Lipschitz Functions Preface”: Cobzas, S Miculescu, R Nicolae, A, Lipschitz Functions, Lect. Notes Math., Lecture Notes in Mathematics, 2241, Springer International Publishing Ag, 2019, V+  isi
    70. Li Ch., Ouyang J., Li X., “Classifying Extremely Short Texts By Exploiting Semantic Centroids in Word Mover'S Distance Space”, Web Conference 2019: Proceedings of the World Wide Web Conference (Www 2019), Assoc Computing Machinery, 2019, 939–949  crossref  isi
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    72. Dello Schiavo L., “Characteristic Functionals of Dirichlet Measures”, Electron. J. Probab., 24 (2019), 115  crossref  isi
    73. Bogachev I V., “Distributions of Polynomials in Many Variables and Nikolskii-Besov Spaces”, Real Anal. Exch., 44:1 (2019), 49–64  crossref  isi
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    76. Alexander V. Kolesnikov, “Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem”, Mosc. Math. J., 20:1 (2020), 67–91  mathnet  crossref
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