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

V. I. Bogachev ; A. V. Kolesnikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 5, pp. 785-890

Résumé

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.

Détail

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

Bibliographie

[1] L. V. Kantorovich, “O peremeschenii mass”, Dokl. AN SSSR, 37 (1942), 227–229 | MR | Zbl

[2] L. V. Kantorovich, “Ob odnoi probleme Monzha”, UMN, 3:2(24) (1948), 225–226

[3] L. V. Kantorovich, G. Sh. Rubinshtein, “Ob odnom funktsionalnom prostranstve i nekotorykh ekstremalnykh zadachakh”, Dokl. AN SSSR, 115:6 (1957), 1058–1061 | MR | Zbl

[4] L. V. Kantorovich, G. Sh. Rubinshtein, “Ob odnom prostranstve vpolne additivnykh funktsii mnozhestva”, Vestn. LGU, 13:7 (1958), 52–59 | MR | Zbl

[5] A. M. Vershik, “The Kantorovich metric: initial history and little-known applications”, J. Math. Sci. (N. Y.), 133:4 (2006), 1410–1417 | DOI | MR | Zbl

[6] G. Monge, Mémoire sur la théorie des déblais et de remblais, Histoire de l'Académie Royale des Sciences avec les Mémoires de Mathématique de Physique, Paris, 1781, 666–704

[7] Leonid Vitalevich Kantorovich: chelovek i uchenyi, v. 1, 2, Nauka Sibiri v litsakh, eds. V. L. Kantorovich, S. S. Kutateladze, Ya. I. Fet, Izd-vo SO RAN, filial “Geo”, Novosibirsk, 2002, 543, 614 pp. | MR | MR | Zbl

[8] A. M. Vershik, S. S. Kutateladze, S. P. Novikov, “Leonid Vital'evich Kantorovich (on the 100th anniversary of his birth)”, Russian Math. Surveys, 67:3 (2012), 589–597 | DOI | DOI

[9] L. Ambrosio, “Lecture notes on optimal transport problems”, Mathematical aspects of evolving interfaces (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, 2003, 1–52 | DOI | MR | Zbl

[10] L. Ambrosio, N. Gigli, A user's guide to optimal transport, Scuola Normale Superiore di Pisa, Pisa, 2011

[11] L. Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures Math. ETH Zurich, Birkhäuser Verlag, Basel, 2005, viii+333 pp. | MR | Zbl

[12] L. C. Evans, “Partial differential equations and Monge–Kantorovich mass transfer”, Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, 65–126 | MR | Zbl

[13] W. Gangbo, R. J. McCann, “The geometry of optimal transportation”, Acta Math., 177:2 (1996), 113–161 | DOI | MR | Zbl

[14] R. J. McCann, N. Guillen, Five lectures on optimal transportation: geometry, regularity and applications, 2010, arXiv: 1011.2911v1

[15] S. T. Rachev, L. Rüschendorf, Mass transportation problems, v. I, II, Probab. Appl. (N. Y.), Springer-Verlag, New York, 1998, xxvi+508, xxvi+430 pp. | MR | MR | Zbl

[16] C. Villani, Topics in optimal transportation, Grad. Stud. Math., 58, Amer. Math. Soc., Providence, RI, 2003, xvi+370 pp. | MR | Zbl

[17] C. Villani, Optimal transport. Old and new, Grundlehren Math. Wiss., 338, Springer-Verlag, Berlin, 2009, xxii+973 pp. | DOI | MR | Zbl

[18] A. M. Vershik, “Some remarks on the infinite-dimensional problems of linear programming”, Russian Math. Surveys, 25:5 (1970), 117–124 | DOI | MR | Zbl

[19] A. N. Bogolyubov, Gaspar Monzh, Nauka, M., 1978, 184 pp.

[20] P. Appell, “Mémoire sur les déblais et les remblais des systèmes continus ou discontinus”, Mém. Acad. Sci. Inst. France, 29, Paris, 1887, 208 pp.

[21] V. N. Sudakov, “Geometric problems of the theory of infinite-dimensional probability distributions”, Proc. Steklov Inst. Math., 141 (1979), 1–178 | MR | Zbl

[22] L. Ambrosio, B. Kirchheim, A. Pratelli, “Existence of optimal transport maps for chrystalline norms”, Duke Math. J., 125:2 (2004), 207–241 | DOI | MR | Zbl

[23] L. C. Evans, W. Gangbo, Differential equations methods for the Monge–Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137, No 653, 1999, viii+66 pp. | MR | Zbl

[24] T. Champion, L. De Pascale, “The Monge problem in $\mathbb{R}^d$”, Duke Math. J., 157:3 (2011), 551–572 | DOI | MR | Zbl

[25] V. I. Bogachev, “O rabotakh G. M. Fikhtengoltsa po teorii integrala”, Istor.-\allowbreakmatem. issled. Ser. 2, 2005, no. 9(44), 252–264 | MR | Zbl

[26] L. V. Kantorovich, Matematicheskie metody organizatsii i planirovaniya proizvodstva, Izd-vo LGU, L., 1939, 67 pp.; �зд. дом СПбГУ, СПб., 2012

[27] A. M. Vershik, “Many-valued measure-preserving mappings (polymorphisms) and Markovian operators”, J. Soviet Math., 23:3 (1983), 2243–2266 | DOI | MR | Zbl

[28] V. L. Levin, “O teoremakh dvoistvennosti v zadache Monzha–Kantorovicha”, UMN, 32:3(195) (1977), 171–172 | MR | Zbl

[29] V. I. Bogachev, Measure theory, v. I, II, Springer-Verlag, Berlin, 2007, xviii+500, xiv+575 pp. | MR | Zbl

[30] R. Fortet, E. Mourier, “Convergence de la répartition empirique vers la répartition théorique”, Ann. Sci. Ecole Norm. Sup. (3), 70:3 (1953), 267–285 | MR | Zbl

[31] R. F. Arens, J. Eells, Jr., “On embedding uniform and topological spaces”, Pacif. J. Math., 6 (1956), 397–403 | MR | Zbl

[32] D. A. Edwards, “On the Kantorovich–Rubinshtein theorem”, Expos. Math., 29:4 (2011), 387–398 | DOI | MR | Zbl

[33] A. M. Vershik, “Dynamics of metrics in measure spaces and their asymptotic invariants”, Markov Process. Related Fields, 16:1 (2010), 169–184 | MR | Zbl

[34] J. Melleray, F. V. Petrov, A. M. Vershik, “Linearly rigid metric spaces and the embedding problem”, Fund. Math., 199:2 (2008), 177–194 | DOI | MR | Zbl

[35] V. M. Zolotarev, “On the continuity of stochastic sequences generated by recurrent processes”, Theory Probab. Appl., 20:4 (1976), 819–832 | DOI | MR | Zbl

[36] V. M. Zolotarev, “Metric distances in spaces of random variables and their distributions”, Math. USSR-Sb., 30:3 (1976), 373–401 | DOI | MR | Zbl

[37] S. T. Rachev, “Monge–Kantorovich problem on mass transition and its applications in stochastics”, Theory Probab. Appl., 29:4 (1985), 625–653 | DOI | MR | Zbl

[38] R. L. Dobrushin, “Prescribing a system of random variables by conditional distributions”, Theory Probab. Appl., 15:3 (1970), 458–486 | DOI | MR | Zbl

[39] L. N. Vasershtein, “Markov processes over denumerable products of spaces describing large system of automata”, Problems Inform. Transmission, 5:3 (1969), 47–52 | MR | Zbl

[40] Y. Olivier, “A survey of Ricci curvature for metric spaces and Markov chains”, Probabilistic approach to geometry, Adv. Stud. Pure Math., 57, Math. Soc. Japan, Tokyo, 2010, 343–381 | MR | Zbl

[41] L. Rüschendorf, “On $c$-optimal random variables”, Statist. Probab. Lett., 27:3 (1996), 267–270 | DOI | MR | Zbl

[42] M. Knott, C. S. Smith, “On the optimal mapping of distributions”, J. Optim. Theory Appl., 43:1 (1984), 39–49 | DOI | MR | Zbl

[43] M. Beiglböck, M. Goldstern, G. Maresch, W. Schachermayer, “Optimal and better transport plans”, J. Funct. Anal., 256:6 (2009), 1907–1927 | DOI | MR | Zbl

[44] A. Yu. Plakhov, “Precise solutions of the one-dimensional Monge–Kantorovich problem”, Sb. Math., 195:9 (2004), 1291–1307 | DOI | DOI | MR | Zbl

[45] V. L. Levin, “Optimality conditions and exact solutions to the two-dimensional Monge–Kantorovich problem”, J. Math. Sci. (N. Y.), 133:4 (2006), 1456–1463 | DOI | MR | Zbl

[46] V. L. Levin, “General Monge–Kantorovich problem and its applications in measure theory and mathematical economics”, Functional analysis, optimization, and mathematical economics, A collection of papers dedicated to the memory of L. V. Kantorovich, ed. L. J. Leifman, Oxford Univ. Press, New York, 1990, 141–176 | MR | Zbl

[47] D. Ramachandran, L. Rüschendorf, “A general duality theorem for marginal problems”, Probab. Theory Relat. Fields, 101:3 (1995), 311–319 | DOI | MR | Zbl

[48] D. Ramachandran, L. Rüschendorf, “Duality and perfect probability spaces”, Proc. Amer. Math. Soc., 124:7 (1996), 2223–2228 | DOI | MR | Zbl

[49] D. Ramachandran, L. Rüschendorf, “On the Monge–Kantorovich duality theorem”, Theory Probab. Appl., 45:2 (2001), 350–356 | DOI | MR | Zbl

[50] M. Beiglböck, W. Schachermayer, “Duality for Borel measurable cost functions”, Trans. Amer. Math. Soc., 363:8 (2011), 4203–4224 | DOI | MR | Zbl

[51] H. G. Kellerer, “Duality theorems for marginal problems”, Z. Wahrscheinlichkeitstheor. verw. Geb., 67:4 (1984), 399–432 | DOI | MR | Zbl

[52] D. A. Edwards, “A simple proof in Monge–Kantorovich duality theory”, Studia Math., 200:1 (2010), 67–77 | DOI | MR | Zbl

[53] L. Ambrosio, A. Pratelli, “Existence and stability results in the $L^1$ theory of optimal transportation”, Optimal transportation and applications (Martina Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, 2003, 123–160 | DOI | MR | Zbl

[54] V. L. Levin, “A formula for the optimal value in the Monge–Kantorovich problem with a smooth cost function, and a characterization of cyclically monotone mappings”, Math. USSR-Sb., 71:2 (1992), 533–548 | DOI | MR | Zbl

[55] V. L. Levin, “On duality theory for non-topological variants of the mass transfer problem”, Sb. Math., 188:4 (1997), 571–602 | DOI | DOI | MR | Zbl

[56] V. L. Levin, “Dvoistvennost Monzha–Kantorovicha i ee primenenie v teorii poleznosti”, Ekonomika i matem. metody, 47:4 (2011), 143–165

[57] V. Levin, “Abstract cyclical monotonicity and Monge solutions for the general Monge–Kantorovich problem”, Set-Valued Anal., 7:1 (1999), 7–32 | DOI | MR | Zbl

[58] V. L. Levin, A. A. Milyutin, “The problem of mass transfer with a discontinuous cost function and a mass statement of the duality problem for convex extremal problems”, Russian Math. Surveys, 34:3 (1979), 1–78 | DOI | MR | Zbl

[59] L. A. Caffarelli, M. Feldman, R. J. McCann, “Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs”, J. Amer. Math. Soc., 15:1 (2002), 1–26 | DOI | MR | Zbl

[60] N. S. Trudinger, X.-J. Wang, “On the Monge mass transfer problem”, Calc. Var. Partial Differential Equations, 13:1 (2001), 19–31 | DOI | MR | Zbl

[61] T. Champion, L. De Pascale, “The Monge problem for strictly convex norms in $\mathbb{R}^d$”, J. Eur. Math. Soc., 12:6 (2010), 1355–1369 | DOI | MR | Zbl

[62] L. Caravenna, “A proof of Sudakov theorem with strictly convex norms”, Math. Z., 268:1-2 (2011), 371–407 | DOI | MR | Zbl

[63] A. V. Pogorelov, Mnogomernoe uravnenie Monzha–Ampera $\det\|z_{ij}\|=\phi(z_1,\dots,z_n,z,x_1,\dots,x_n)$, Nauka, M., 1988, 94 pp. | MR | Zbl

[64] C. E. Gutiérrez, The Monge–Ampère equation, Progr. Nonlinear Differential Equations Appl., 44, Birkhäuser Boston, Inc., Boston, MA, 2001, xii+127 pp. | MR | Zbl

[65] Y. Brenier, “Polar factorization and monotone rearrangement of vector valued functions”, Comm. Pure Appl. Math., 44:4 (1991), 375–417 | DOI | MR | Zbl

[66] R. J. McCann, “Existence and uniqueness of monotone measure-preserving maps”, Duke Math. J., 80:2 (1995), 309–323 | DOI | MR | Zbl

[67] V. I. Bogachev, Differentiable measures and the Malliavin calculus, Math. Surveys Monogr., 164, Amer. Math. Soc., Providence, RI, 2010, xvi+488 pp. | MR | Zbl

[68] R. J. McCann, “A convexity principle for interacting gases”, Adv. Math., 128:1 (1997), 153–179 | DOI | MR | Zbl

[69] R. van der Putten, “Uniqueness and non uniqueness of optimal maps in mass transport problem with not strictly convex cost”, Comment. Math. Univ. Carolin., 51:1 (2010), 67–83 | MR | Zbl

[70] N. V. Krylov, “Fully nonlinear second order elliptic equations: recent development”, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25:3-4 (1997), 569–595 | MR | Zbl

[71] X. Cabré, “Elliptic PDE's in probability and geometry: symmetry and regularity of solutions”, Discrete Contin. Dyn. Syst., 20:3 (2008), 425–457 | DOI | MR | Zbl

[72] N. S. Trudinger, X.-J. Wang, “The Monge–Ampère equation and its geometric applications”, Handbook of geometric analysis, No 1, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008, 467–524 | MR | Zbl

[73] L. Forzani, D. Maldonaldo, “Properties of the solutions to the Monge–Ampère equation”, Nonlinear Anal., 57:5-6 (2004), 815–829 | DOI | MR | Zbl

[74] G. De Philippis, A. Figalli, $W^{2,1}$-regularity for solutions of the Monge–Ampère equation, 2011, arXiv: 1111.7207v2

[75] I. Fragalà, M. S. Gelli, A. Pratelli, “Continuity of an optimal transport in Monge problem”, J. Math. Pures Appl. (9), 84:9 (2005), 1261–1294 | DOI | MR | Zbl

[76] X.-J. Wang, “Some counterexamples to the regularity of Monge–Ampère equations”, Proc. Amer. Math. Soc., 123:3 (1995), 841–845 | MR | Zbl

[77] L. A. Caffarelli, “Monotonicity properties of optimal transportation and the FKG and related inequalities”, Comm. Math. Phys., 214:3 (2000), 547–563 | DOI | MR | Zbl

[78] D. Bakry, M. Ledoux, “Lévy–Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator”, Invent. Math., 123:2 (1996), 259–281 | DOI | MR | Zbl

[79] D. Cordero-Erausquin, “Some applications of mass transport to Gaussian-type inequalities”, Arch. Ration. Mech. Anal., 161:3 (2002), 257–269 | DOI | MR | Zbl

[80] A. V. Kolesnikov, “Global Hölder estimates for optimal transportation”, Math. Notes, 88:5-6 (2010), 678–695 | DOI | DOI | MR | Zbl

[81] B. Klartag, Poincaré inequalities and moment maps, 2011, arXiv: 1104.2791

[82] A. V. Kolesnikov, “Sobolevskaya regulyarnost transportirovki veroyatnostnykh mer i transportnye neravenstva”, Teoriya veroyatn. i ee primen., 57:2 (2012), 296–321

[83] V. I. Bogachev, A. V. Kolesnikov, “Sobolevskaya regulyarnost dlya beskonechnomernogo uravneniya Monzha–Ampera”, Dokl. RAN, 444:2 (2012), 131–136

[84] V. I. Bogachev, A. V. Kolesnikov, “Sobolev regularity for the Monge–Ampere equation in the Wiener space”, Kyoto J. Math. (to appear); 2011, arXiv: 1110.1822

[85] L. De Pascale, L. C. Evans, A. Pratelli, “Integral estimates for transport densities”, Bull. London Math. Soc., 36:3 (2004), 383–395 | DOI | MR | Zbl

[86] G. Loeper, “On the regularity of solutions of optimal transportation problems”, Acta Math., 202:2 (2009), 241–283 | DOI | MR | Zbl

[87] J. Liu, N. S. Trudinger, X.-J. Wang, “Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation”, Comm. Partial Differential Equations, 35:1 (2010), 165–184 | DOI | MR | Zbl

[88] G. Loeper, C. Villani, “Regularity of optimal transport in curved geometry: the nonfocal case”, Duke Math. J., 151:3 (2010), 431–485 | DOI | MR | Zbl

[89] L. Caffarelli, R. J. McCann, “Free boundaries in optimal transport and Monge–Ampère obstacle problems”, Ann. of Math. (2), 171:2 (2010), 673–730 | DOI | MR | Zbl

[90] A. Figalli, “The optimal partial transport problem”, Arch. Ration. Mech. Anal., 195:2 (2010), 533–560 | DOI | MR | Zbl

[91] A. Figalli, Y.-H. Kim, “Partial regularity of Brenier solutions of the Monge–Ampère equation”, Discrete Contin. Dyn. Syst., 28:2 (2010), 559–565 | DOI | MR | Zbl

[92] A. A. Lipchius, “A note on the equality in the Monge and Kantorovich problems”, Theory Probab. Appl., 50:4 (2006), 689–693 | DOI | DOI | MR | Zbl

[93] A. Pratelli, “On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation”, Ann. Inst. H. Poincaré Probab. Statist., 43:1 (2007), 1–13 | DOI | MR | Zbl

[94] V. D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, with an appendix by M. Gromov, Lecture Notes in Math., 1200, Springer-Verlag, Berlin, 1986, viii+156 pp. | MR | Zbl

[95] D. Cordero-Erausquin, B. Nazaret, C. Villani, “A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities”, Adv. Math., 182:2 (2004), 307–332 | DOI | MR | Zbl

[96] B. Nazaret, “Best constant in Sobolev trace inequalities on the half space”, Nonlinear Anal., 65:10 (2006), 1977–1985 | DOI | MR | Zbl

[97] F. Maggi, C. Villani, “Balls have the worst best Sobolev inequalities”, J. Geom. Anal., 15:1 (2005), 83–121 ; “Balls have the worst best Sobolev inequalities. II: Variants and extensions”, Calc. Var. Partial Differential Equations, 31:1 (2008), 47–74 | DOI | MR | Zbl | DOI | MR | Zbl

[98] A. Figalli, F. Maggi, A. Pratelli, “A mass transportation approach to quantitative isoperimetric inequalities”, Invent. Math., 182:1 (2010), 167–211 | DOI | MR | Zbl

[99] D. Cordero-Erausquin, R. J. McCann, M. Schmuckenschläger, “A Riemannian interpolation inequality à la Borell, Brascamp and Lieb”, Invent. Math., 146:2 (2001), 219–257 | DOI | MR | Zbl

[100] D. Cordero-Erausquin, R. J. McCann, M. Schmuckenschläger, “Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport”, Ann. Fac. Sci. Toulouse Math. (6), 15:4 (2006), 613–635 | DOI | MR | Zbl

[101] P. Castillon, “Submanifolds, isoperimetric inequalities and optimal transportation”, J. Funct. Anal., 259:1 (2010), 79–103 ; (2009), arXiv: 0908.2711 | DOI | MR | Zbl

[102] L. Gross, “Logarithmic Sobolev inequalities”, Amer. J. Math., 97:4 (1975), 1061–1083 | DOI | MR | Zbl

[103] M. Ledoux, The concentration of measure phenomenon, Math. Surveys Monogr., 89, Amer. Math. Soc., Providence, RI, 2001, x+181 pp. | MR | Zbl

[104] V. I. Bogachev, Gaussovskie mery, Nauka, M., 1997, 352 pp. | MR | Zbl

[105] S. G. Bobkov, F. Götze, “Exponential integrability and transportation cost related to logarithmic Sobolev inequality”, J. Funct. Anal., 163 (1999), 1–28 | DOI | MR | Zbl

[106] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, 2002, arXiv: math/0211159 | Zbl

[107] V. N. Sudakov, B. S. Tsirelson, “Ekstremalnye svoistva poluprostranstv dlya sfericheski invariantnykh mer”, Problemy teorii veroyatnostnykh raspredelenii. II, Zap. nauchn. sem. LOMI, 41, Nauka, L., 1974, 14–24 | MR | Zbl

[108] C. Borell, “The Brunn–Minkowski inequality in Gauss space”, Invent. Math., 30:2 (1975), 207–216 | DOI | MR | Zbl

[109] F.-Y. Wang, “Logarithmic Sobolev inequalities on non-compact Riemannian manifolds”, Probab. Theory Related Fields, 109:3 (1997), 417–424 | DOI | MR | Zbl

[110] F. Barthe, A. V. Kolesnikov, “Mass transport and variants of the logarithmic Sobolev inequality”, J. Geom. Anal., 18:4 (2008), 921–979 | DOI | MR | Zbl

[111] O. S. Rothaus, “Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities”, J. Funct. Anal., 64:2 (1985), 296–313 | DOI | MR | Zbl

[112] F. Barthe, P. Cattiaux, C. Roberto, “Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry”, Rev. Mat. Iberoam., 22:3 (2006), 993–1067 | DOI | MR | Zbl

[113] D. Cordero-Erausquin, W. Gangbo, C. Houdré, “Inequalities for generalized entropies and optimal transportation”, Recent advances in the theory and applications of mass transport, Contemp. Math., 353, Amer. Math. Soc., Providence, RI, 2004, 73–94 | DOI | MR | Zbl

[114] E. Milman, S. Sodin, “An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies”, J. Funct. Anal., 254:5 (2008), 1235–1268 | DOI | MR | Zbl

[115] S. G. Bobkov, “Isoperimetric and analytic inequalities for log-concave probability measures”, Ann. Probab., 27:4 (1999), 1903–1921 | DOI | MR | Zbl

[116] A. V. Kolesnikov, “Modified log-Sobolev inequalities and isoperimetry”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18:2 (2007), 179–208 | DOI | MR | Zbl

[117] M. Talagrand, “Transportation cost for Gaussian and other product measures”, Geom. Funct. Anal., 6:3 (1996), 587–600 | DOI | MR | Zbl

[118] V. I. Bogachev, A. V. Kolesnikov, “Integrability of absolutely continuous transformations of measures and applications to optimal mass transportation”, Theory Probab. Appl., 50:3 (2006), 367–385 | DOI | DOI | MR | Zbl

[119] F. Otto, C. Villani, “Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality”, J. Funct. Anal., 173:2 (2000), 361–400 | DOI | MR | Zbl

[120] S. G. Bobkov, I. Gentil, M. Ledoux, “Hypercontractivity of Hamilton–Jacobi equations”, J. Math. Pures Appl. (9), 80:7 (2001), 669–696 | DOI | MR | Zbl

[121] P. Cattiaux, A. Guillin, “On quadratic transportation cost inequalities”, J. Math. Pures Appl. (9), 86:4 (2006), 342–361 | DOI | MR | Zbl

[122] N. Gozlan, “Characterization of Talagrand's like transportation-cost inequalities on the real line”, J. Funct. Anal., 250:2 (2007), 400–425 | DOI | MR | Zbl

[123] N. Gozlan, “A characterization of dimension free concentration in terms of transportation inequalities”, Ann. Probab., 37:6 (2009), 2480–2498 | DOI | MR | Zbl

[124] I. Gentil, A. Guillin, L. Miclo, “Modified logarithmic Sobolev inequalities and transportation inequalities”, Probab. Theory Related Fields, 133:3 (2005), 409–436 | DOI | MR | Zbl

[125] H. Djellout, A. Guillin, L. Wu, “Transportation cost-information inequalities and applications to random dynamical systems and diffusions”, Ann. Probab., 32:3B (2004), 2702–2732 | DOI | MR | Zbl

[126] F. Bolley, C. Villani, “Weighted Csiszár–Kullback–Pinsker inequalities and application to transportation inequalities”, Ann. Fac. Sci. Toulouse Math. (6), 14:3 (2005), 331–352 | DOI | MR | Zbl

[127] N. Gozlan, C. Roberto, P. M. Samson, “A new characterization of Talagrand's transport-entropy inequalities and applications”, Ann. Probab., 39:3 (2011), 857–880 | DOI | MR | Zbl

[128] P. Cattiaux, A. Guillin, L.-M. Wu, “A note on Talagrand's transportation inequality and logarithmic Sobolev inequality”, Probab. Theory Related Fields, 148:1-2 (2010), 285–304 | DOI | MR | Zbl

[129] A. Guillin, C. Léonard, L. Wu, N. Yao, “Transportation-information inequalities for Markov processes”, Probab. Theory Related Fields, 144:3-4 (2009), 669–695 | DOI | MR | Zbl

[130] A. Guillin, C. Leonard, F.-Y. Wang, L. Wu, Transportation-information inequalities for Markov processes (II): relations with other functional inequalities, 2009, arXiv: 0902.2101

[131] N. Gozlan, C. Léonard, “Transport inequalities. A survey”, Markov Process. Related Fields, 16:4 (2010), 635–736 | MR | Zbl

[132] J. Dolbeault, B. Nazaret, G. Savaré, “A new class of transport distances between measures”, Calc. Var. Partial Differential Equations, 34:2 (2009), 193–231 | DOI | MR | Zbl

[133] L. Gozlan, C. Léonard, “A large deviation approach to some transportation cost inequalities”, Probab. Theory Related Fields, 139:1-2 (2007), 235–283 | DOI | MR | Zbl

[134] E. Milman, “Isoperimetric and concentration inequalities: equivalence under curvature lower bound”, Duke Math. J., 154:2 (2010), 207–239 | DOI | MR | Zbl

[135] E. Milman, “Properties of isoperimetric, functional and transport-entropy inequalities via concentration”, Probab. Theory Related Fields, 152:3-4 (2012), 475–507 | DOI | MR | Zbl

[136] J. Lott, C. Villani, “Ricci curvature for metric-measure spaces via optimal transport”, Ann. Math. (2), 169:3 (2009), 903–991 | DOI | MR | Zbl

[137] L. Ambrosio, A. Figalli, “Geodesics in the space of measure-preserving maps and plans”, Arch. Ration. Mech. Anal., 194:2 (2009), 421–462 | DOI | MR | Zbl

[138] S. Angenent, S. Haker, A. Tannenbaum, “Minimizing flows for the Monge–Kantorovich problem”, SIAM J. Math. Anal., 35:1 (2003), 61–97 | DOI | MR | Zbl

[139] L. De Pascale, M. S. Gelli, L. Granieri, “Minimal measures, one-dimensional currents and the Monge–Kantorovich problem”, Calc. Var. Partial Differential Equations, 27:1 (2006), 1–23 | DOI | MR | Zbl

[140] A. Figalli, N. Gigli, “A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions”, J. Math. Pures Appl. (9), 94:2 (2010), 107–130 | DOI | MR | Zbl

[141] J.-D. Benamou, Y. Brenier, “A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem”, Numer. Math., 84:3 (2000), 375–393 | DOI | MR | Zbl

[142] N. Juillet, “On displacement interpolation of measures involved in Brenier's theorem”, Proc. Amer. Math. Soc., 139:10 (2011), 3623–3632 | DOI | MR | Zbl

[143] K.-Th. Sturm, “On the geometry of metric measure spaces, I, II”, Acta Math., 196 (2006), 65–131, 133–177 | DOI | MR | Zbl

[144] J. Lott, “Some geometric calculations on Wasserstein spaces”, Comm. Math. Phys., 277:2 (2008), 423–437 | DOI | MR | Zbl

[145] F. Otto, “The geometry of dissipative evolution equations: the porous medium equation”, Comm. Partial Differential Equations, 26:1-2 (2001), 101–174 | DOI | MR | Zbl

[146] F. Otto, M. Westdickenberg, “Eulerian calculus for the contraction in the Wasserstein distance”, SIAM J. Math. Anal., 37:4 (2005), 1227–1255 (electronic) | DOI | MR | Zbl

[147] U. Gianazza, G. Savaré, G. Toscani, “The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation”, Arch. Ration. Mech. Anal., 194:1 (2009), 133–220 | DOI | MR | Zbl

[148] R. L. Dobrushin, “Vlasov equations”, Funct. Anal. Appl., 13:2 (1979), 115–123 | DOI | MR | Zbl

[149] R. Jordan, D. Kinderlehrer, F. Otto, “The variational formulation of the Fokker–Planck equation”, SIAM J. Math. Anal., 29:1 (1998), 1–17 | DOI | MR | Zbl

[150] L. Natile, M. A. Peletier, G. Savaré, “Contraction of general transportation costs along solutions to Fokker–Planck equations with monotone drifts”, J. Math. Pures Appl. (9), 95:1 (2011), 18–35 | DOI | MR | Zbl

[151] K. Kuwada, “Duality on gradient estimates and Wasserstein controls”, J. Funct. Anal., 258:11 (2010), 3758–3774 | DOI | MR | Zbl

[152] M.-K. von Renesse, K.-Th. Sturm, “Entropic measure and Wasserstein diffusion”, Ann. Probab., 37:3 (2009), 1114–1191 | DOI | MR | Zbl

[153] A. D. Aleksandrov, Izbrannye trudy, v. 1, 2, Nauka, Novosibirsk, 2006, 2007, lii+748, iv+491 pp. | MR | Zbl

[154] V. Oliker, “Embeddings of $S^n$ into $\mathbb{R}^{n+1}$ with given integral Gauss curvature and optimal mass transport on $S^n$”, Adv. Math., 213:2 (2007), 600–620 | DOI | MR | Zbl

[155] R. J. McCann, “Polar factorization of maps on Riemannian manifolds”, Geom. Funct. Anal., 11:3 (2001), 589–608 | DOI | MR | Zbl

[156] A. Fathi, A. Figalli, “Optimal transportation on non-compact manifolds”, Israel J. Math., 175 (2010), 1–59 | DOI | MR | Zbl

[157] A. Figalli, L. Rifford, “Mass transportation on sub-Riemannian manifolds”, Geom. Funct. Anal., 20:1 (2010), 124–159 | DOI | MR | Zbl

[158] Y.-H. Kim, R. J. McCann, “Continuity, curvature, and the general covariance of optimal transportation”, J. Eur. Math. Soc., 12:4 (2010), 1009–1040 | DOI | MR | Zbl

[159] J.-P. Bourguignon, “Ricci curvature and measures”, Japan. J. Math., 4:1 (2009), 27–45 | DOI | MR | Zbl

[160] M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Math., 1, eds. J. Lafontaine, P. Pansu, CEDIC, Paris, 1981, iv+152 pp. | MR | MR | Zbl | Zbl

[161] A. M. Vershik, “The universal Urysohn space, Gromov metric triples and random metrics on the natural numbers”, Russian Math. Surveys, 53:5 (1998), 921–928 | DOI | DOI | MR | Zbl

[162] A. M. Vershik, “Random metric spaces and universality”, Russian Math. Surveys, 59:2 (2004), 259–295 | DOI | DOI | MR | Zbl

[163] K. Bacher, “On Borell–Brascamp–Lieb inequalities on metric measure spaces”, Potential Anal., 33:1 (2010), 1–15 | DOI | MR | Zbl

[164] K. Bacher, K.-Th. Sturm, “Localization and tensorization properties of the curvature-dimension condition for metric measure spaces”, J. Funct. Anal., 259:1 (2010), 28–56 | DOI | MR | Zbl

[165] Q. Deng, K.-Th. Sturm, “Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, II”, J. Funct. Anal., 260:12 (2011), 3718–3725 | DOI | MR | Zbl

[166] D. Bakry, M. Émery, “Diffusions hypercontractives”, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., 1123, Springer, Berlin, 1985, 177–206 | DOI | MR | Zbl

[167] V. I. Bogachev, A. V. Kolesnikov, “Mass transport generated by a flow of Gauss maps”, J. Funct. Anal., 256:3 (2009), 940–957 | DOI | MR | Zbl

[168] G. Huisken, “Flow by mean curvature of convex surfaces into spheres”, J. Differential Geom., 20:1 (1984), 237–266 | MR | Zbl

[169] K. Tso, “Deforming a hypersurfaces by its Gauss–Kronecker curvature”, Comm. Pure Appl. Math., 38:6 (1985), 867–882 | DOI | MR | Zbl

[170] A. V. Kolesnikov, Weak regularity of Gauss mass transport, 2009, arXiv: 0904.1852

[171] Y.-H. Kim, J. Streets, M. Warren, Parabolic optimal transport equations on manifolds, 2010, arXiv: 1008.3824v1

[172] R. J. McCann, P. M. Topping, “Ricci flow, entropy and optimal transportation”, Amer. J. Math., 132:3 (2010), 711–730 | DOI | MR | Zbl

[173] P. Topping, Ricci flows: the foundations via optimal transportation http://homepages.warwick.ac.uk/~maseq/grenoble_20100324.pdf

[174] P. Topping, “$\mathscr{L}$-optimal transportation for Ricci flows”, J. Reine Angew. Math., 636 (2009), 93–122 | DOI | MR | Zbl

[175] J. Lott, “Optimal transport and Perelman's reduced volume”, Calc. Var. Partial Differential Equations, 36:1 (2009), 49–84 | DOI | MR | Zbl

[176] G. Buttazzo, A. Pratelli, S. Solimini, E. Stepanov, Optimal urban networks via mass transportation, Lecture Notes in Math., 1961, Springer-Verlag, Berlin, 2009, x+150 pp. | DOI | MR | Zbl

[177] M. Bernot, V. Caselles, J.-M. Morel, Optimal transportation networks. Models and theory, Lecture Notes in Math., 1955, Springer-Verlag, Berlin, 2009, x+200 pp. | MR | Zbl

[178] A. S. Üstünel, M. Zakai, Transformation of measure on Wiener space, Springer Monogr. Math., Springer-Verlag, Berlin, 2000, xiv+296 pp. | MR | Zbl

[179] V. I. Bogachev, A. V. Kolesnikov, K. V. Medvedev, “Triangular transformations of measures”, Sb. Math., 196:3-4 (2005), 309–335 | DOI | DOI | MR | Zbl

[180] D. Feyel, A. S. Üstünel, “Monge–Kantorovitch measure transportation and Monge–Ampère equation on Wiener space”, Probab. Theory Related Fields, 128:3 (2004), 347–385 | DOI | MR | Zbl

[181] D. Feyel, A. S. Üstünel, “Solution of the Monge–Ampère equation on Wiener space for general log-concave measures”, J. Funct. Anal., 232:1 (2006), 29–55 | DOI | MR | Zbl

[182] D. Feyel, A. S. Üstünel, “The notions of convexity and concavity on Wiener space”, J. Funct. Anal., 176:2 (2000), 400–428 | DOI | MR | Zbl

[183] A. V. Kolesnikov, “Convexity inequalities and optimal transport of infinite-dimensional measures”, J. Math. Pures Appl. (9), 83:11 (2004), 1373–1404 | DOI | MR | Zbl

[184] F. Cavalletti, “The Monge problem in Wiener space”, Calc. Var. Partial Differential Equations, 45:1-2 (2012), 101–124 ; (2011), arXiv: 1103.2798v2 | DOI | MR

[185] V. I. Bogachev, A. V. Kolesnikov, “On the Monge–Ampère equation in infinite dimensions”, Infin. Dimen. Anal. Quantum Probab. Related Top., 8:4 (2005), 547–572 | DOI | MR | Zbl

[186] S. Fang, J. Shao, K.-Th. Sturm, “Wasserstein space over the Wiener space”, Probab. Theory Related Fields, 146:3-4 (2010), 535–565 | DOI | MR | Zbl

[187] L. Decreusefond, “Wasserstein distance on configuration space”, Potential Anal., 28:3 (2008), 283–300 | DOI | MR | Zbl

[188] V. I. Bogachev, A. V. Kolesnikov, “Nonlinear transformations of convex measures”, Theory Probab. Appl., 50:1 (2006), 34–52 | DOI | DOI | MR | Zbl

[189] S. G. Bobkov, “Large deviations via transference plans”, Advances in mathematics research, v. 2, Adv. Math. Res., 2, Nova Sci. Publ., Hauppauge, NY, 2003, 151–175 | MR | Zbl

[190] G. Carlier, A. Galichon, F. Santambrogio, “From Knothe's transport to Brenier's map and a continuation method for optimal transport”, SIAM J. Math. Anal., 41:6 (2009/10), 2554–2576 | DOI | MR | Zbl

[191] J. Moser, “On the volume elements on a manifold”, Trans. Amer. Math. Soc., 120 (1965), 286–294 | DOI | MR | Zbl

[192] Y.-H. Kim and E. Milman, “A generalization of Caffarelli's contraction theorem via (reverse) heat flow”, Math. Ann., 354:3 (2012), 827–862

[193] J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, D. Slepčev, “Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations”, Duke Math. J., 156:2 (2011), 229–271 | DOI | MR | Zbl

[194] P. Bernard, “Young measures, superposition and transport”, Indiana Univ. Math. J., 57:1 (2008), 247–275 | DOI | MR | Zbl

[195] W. Gangbo, A. Świȩch, “Optimal maps for the multidimensional Monge–Kantorovich problem”, Comm. Pure Appl. Math., 51:1 (1998), 23–45 | 3.0.CO;2-H class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[196] W. Gangbo, H. K. Kim, T. Pacini, Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems, Mem. Amer. Math. Soc., 211, No 993, 2011, vi+77 pp. | DOI | MR | Zbl