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

RUS ENG

JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PERSONAL OFFICE

JavaScript is disabled in your browser. Please switch it on to enable full functionality of the website

	General information
	Latest issue
	Archive
	Impact factor
	Subscription
	License agreement
	Submit a manuscript

	Search papers
	Search references

	RSS
	Latest issue
	Current issues
	Archive issues
	What is RSS

Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
	Find

Personal entry:
Login:
Password:
	Save password
	Enter
	Forgotten password?
	Register

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

This article is cited in 60 scientific papers (total in 60 papers)

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

V. I. Bogachev^ab, A. V. Kolesnikov^c

^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.

DOI: https://doi.org/10.4213/rm9490

Full text: PDF file (1370 kB)

References: PDF file HTML file

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

Bibliographic databases:

Document Type: Article
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

\Bibitem{BogKol12}

\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

\mathnet{http://mi.mathnet.ru/umn9490}

\crossref{https://doi.org/10.4213/rm9490}

\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3058744}

\zmath{https://zbmath.org/?q=an:06148569}

\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2012RuMaS..67..785B}

\elib{http://elibrary.ru/item.asp?id=20423462}

\transl

\jour Russian Math. Surveys

\yr 2012

\vol 67

\issue 5

\pages 785--890

\crossref{https://doi.org/10.1070/RM2012v067n05ABEH004808}

\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000314222000001}

\elib{http://elibrary.ru/item.asp?id=20485666}

\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84872294543}

Linking options:

http://mi.mathnet.ru/eng/umn9490

https://doi.org/10.4213/rm9490

http://mi.mathnet.ru/eng/umn/v67/i5/p3

SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:

A. M. Vershik, “Long history of the Monge-Kantorovich transportation problem”, Math. Intelligencer, 35:4 (2013), 1–9
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
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
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
A. M. Vershik, “Two ways to define compatible metrics on the simplex of measures”, J. Math. Sci. (N. Y.), 196:2 (2014), 138–143
A. V. Kolesnikov, S. Yu. Tikhonov, “Regularity of the Monge–Ampère equation in Besov's spaces”, Calc. Var., 49:3-4 (2014), 1187–1197
A. V. Kolesnikov, M. Röckner, “On continuity equations in infinite dimensions with non-Gaussian reference measure”, J. Funct. Anal., 266:7 (2014), 4490–4537
O. Butkovsky, “Subgeometric rates of convergence of Markov processes in the Wasserstein metric”, Ann. Appl. Probab., 24:2 (2014), 526–552
S. Caracciolo, C. Lucibello, G. Parisi, G. Sicuro, “Scaling hypothesis for the Euclidean bipartite matching problem”, Phys. Rev. E, 90:1 (2014), 012118
D. B. Bukin, “On the Monge and Kantorovich problems for distributions of diffusion processes”, Math. Notes, 96:5-6 (2014), 864–870
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
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
V. I. Bogachev, A. V. Shaposhnikov, “Lower bounds for the Kantorovich distance”, Dokl. Math., 91:1 (2015), 91–93
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
D. Zaev, “On the Monge–Kantorovich Problem with Additional Linear Constraints”, Math. Notes, 98:5 (2015), 725–741
D. A. Zaev, “On ergodic decompositions related to the Kantorovich problem”, J. Math. Sci. (N. Y.), 216:1 (2016), 65–83
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
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
V. I. Bogachev, F.-Y. Wang, A. V. Shaposhnikov, “Estimates of the Kantorovich Norm on Manifolds”, Dokl. Math., 92:1 (2015), 494–499
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
Alexander V. Kolesnikov, Danila A. Zaev, “Exchangeable optimal transportation and log-concavity”, Theory Stoch. Process., 20(36):2 (2015), 54–62
V. I. Bogachev, “Distributions of polynomials on multidimensional and infinite-dimensional spaces with measures”, Russian Math. Surveys, 71:4 (2016), 703–749
D. B. Bukin, “On the Kantorovich Problem for Nonlinear Images of the Wiener Measure”, Math. Notes, 100:5 (2016), 660–665
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
S. G. Bobkov, “Proximity of probability distributions in terms of Fourier–Stieltjes transforms”, Russian Math. Surveys, 71:6 (2016), 1021–1079
V. I. Bogachev, M. Rëckner, S. V. Shaposhnikov, “Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations”, J. Funct. Anal., 271:5 (2016), 1262–1300
A. V. Kolesnikov, E. Milman, “Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities”, Calc. Var. Partial Differential Equations, 55:4 (2016), 77, 36 pp.
V. I. Bogachev, F.-Y. Wang, A. V. Shaposhnikov, “On inequalities relating the Sobolev and Kantorovich norms”, Dokl. Math., 93:3 (2016), 256–258
V. I. Bogachev, M. Rëckner, S. V. Shaposhnikov, “Estimates of distances between transition probabilities of diffusions”, Dokl. Math., 93:2 (2016), 135–139
A. Winter, “Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints”, Comm. Math. Phys., 347:1 (2016), 291–313
I. I. Malofeev, “Measurable dependence of conditional measures on a parameter”, Dokl. Math., 94:2 (2016), 493–497
A. V. Kolesnikov, E. Milman, “Riemannian metrics in $\mathbb R^n$ and Sobolev-type inequalities”, Dokl. Math., 94:2 (2016), 510–513
Alexander V. Kolesnikov, Nikolay Lysenko, “Remarks on mass transportation minimizing expectation of a minimum of affine functions”, Theory Stoch. Process., 21(37):2 (2016), 22–28
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
I. Oliver, Y. Miche, “On the development of a metric for quality of information content over anonymised data-sets”, Proceedings 2016 10Th International Conference on the Quality of Information and Communications Technology (Quatic), IEEE, 2016, 185–190
V. I. Bogachev, A. I. Kirillov, S. V. Shaposhnikov, “Distances between stationary distributions of diffusions and solvability of nonlinear Fokker–Planck–Kolmogorov equations”, Theory Probab. Appl., 62:1 (2018), 12–34
B. Klartag, A. V. Kolesnikov, “Remarks on curvature in the transportation metric”, Anal. Math., 43:1 (2017), 67–88
J. Xie, G. Dai, F. Zhu, Y. Fang, “Learning barycentric representations of 3D shapes for sketch-based 3D shape retrieval”, 30Th IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2017), IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2017, 3615–3623
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
O. A. Manita, “Estimates for transportation costs along solutions to Fokker-Planck-Kolmogorov equations with dissipative drifts”, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28:3 (2017), 601–618
V. I. Bogachev, E. D. Kosov, S. N. Popova, “On Gaussian Nikolskii-Besov classes”, Dokl. Math., 96:2 (2017), 498–502
O. Butkovsky, M. Scheutzow, “Invariant measures for stochastic functional differential equations”, Electron. J. Probab., 22 (2017), 98, 23 pp.
V. I. Bogachev, A. N. Doledenok, S. V. Shaposhnikov, “Weighted Zolotarev metrics and the Kantorovich metric”, Dokl. Math., 95:2 (2017), 113–117
G. Sicuro, “The Euclidean matching problem”, Doctoral thesis accepted by the University of Pisa, Italy, Springer Theses, Springer, Cham, 2017, 1–4
G. Sulyok, S. Sponar, “Heisenberg's error-disturbance uncertainty relation: Experimental study of competing approaches”, Phys. Rev. A, 96:2 (2017), 022137
Nikolay Lysenko, “Maximization of functionals depending on the terminal value and the running maximum of a martingale: a mass transport approach”, Theory Stoch. Process., 22(38):1 (2017), 30–40
Georgii A. Alekseev, Ekaterina V. Yurova, “On Gaussian conditional measures depending on a parameter”, Theory Stoch. Process., 22(38):2 (2017), 1–7
Alexander V. Kolesnikov, Egor D. Kosov, “Moment measures and stability for Gaussian inequalities”, Theory Stoch. Process., 22(38):2 (2017), 47–61
V. I. Bogachev, A. N. Kalinin, S. N. Popova, “O ravenstve znachenii v zadachakh Monzha i Kantorovicha”, Veroyatnost i statistika. 25, Posvyaschaetsya pamyati Vladimira Nikolaevicha SUDAKOVA, Zap. nauchn. sem. POMI, 457, POMI, SPb., 2017, 53–73
S. G. Bobkov, “Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances”, Probab. Theory Relat. Field, 170:1-2 (2018), 229–262
R. S. MacKay, “Management of complex dynamical systems”, Nonlinearity, 31:2 (2018), R52–R65
H. Brezis, “Remarks on the Monge-Kantorovich problem in the discrete setting”, C. R. Math. Acad. Sci. Paris, 356:2 (2018), 207–213
A. N. Doledenok, “On a Kantorovich Problem with a Density Constraint”, Math. Notes, 104:1 (2018), 39–47
V. I. Bogachev, “Ornstein–Uhlenbeck operators and semigroups”, Russian Math. Surveys, 73:2 (2018), 191–260
A. A. Mingazov, D. A. Bykov, L. L. Doskolovich, N. L. Kazanskii, “Variatsionnaya interpretatsiya zadachi rascheta funktsii eikonala iz usloviya formirovaniya zadannogo raspredeleniya osveschennosti”, KO, 42:4 (2018), 568–573
N. L. Kazanskii, S. I. Kharitonov, I. N. Kozlova, M. A. Moiseev, “Svyaz fazovoi problemy v optike, fokusirovki izlucheniya i zadachi Monzha–Kantorovicha”, KO, 42:4 (2018), 574–587
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
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
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
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

Number of views:
This page:	2164
Full text:	616
References:	126
First page:	136

What is a QR-code?

Registration

Logotypes