Optimal Transport: Old and New

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

[1]  Rory A. Fisher,et al.  Theory of Statistical Estimation , 1925, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[3]  Claude E. Shannon,et al.  The Mathematical Theory of Communication , 1950 .

[4]  M. Rosenblatt.  Remarks on a Multivariate Transformation , 1952 .

[5]  H. Knothe.  Contributions to the theory of convex bodies. , 1957 .

[6]  J. Nash.  Continuity of Solutions of Parabolic and Elliptic Equations , 1958 .

[7]  L. Kantorovitch,et al.  On the Translocation of Masses , 1958 .

[8]  A. J. Stam.  Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon , 1959, Inf. Control..

[9]  Amiel Feinstein,et al.  Information and information stability of random variables and processes , 1964 .

[10]  J. Moser.  On the volume elements on a manifold , 1965 .

[11]  L. Kantorovich.  The best use of economic resources , 1967 .

[12]  Edward Nelson.  Derivation of the Schrodinger equation from Newtonian mechanics , 1966 .

[13]  S. Kullback,et al.  A lower bound for discrimination information in terms of variation (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[14]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[15]  H. Fédérer.  Geometric Measure Theory , 1969 .

[16]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 2017 .

[17]  H. Schubert,et al.  O. D. Kellogg, Foundations of Potential Theory. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 31). X + 384 S. m. 30 Fig. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 32,– , 1969 .

[18]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[19]  Brian J. Hoskins,et al.  Atmospheric frontogenesis models: Some solutions , 1971 .

[20]  B. Muckenhoupt.  Hardy's inequality with weights , 1972 .

[21]  C. Mallows.  A Note on Asymptotic Joint Normality , 1972 .

[22]  Hiroshi Tanaka.  An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas , 1973 .

[23]  H. Murata,et al.  An inequality for certain functional of multidimensional probability distributions , 1974 .

[24]  R. Holley.  Remarks on the FKG inequalities , 1974 .

[25]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[26]  L. Gross.  LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .

[27]  R. Dudley.  Probabilities and metrics : convergence of laws on metric spaces, with a view to statistical testing , 1976 .

[28]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[29]  Hiroshi Tanaka.  Probabilistic treatment of the Boltzmann equation of Maxwellian molecules , 1978 .

[30]  R. Osserman.  The isoperimetric inequality , 1978 .

[31]  V. Sudakov,et al.  Geometric Problems in the Theory of Infinite-dimensional Probability Distributions , 1979 .

[32]  R. Greene,et al.  Diffeomorphisms and volume-preserving embeddings of noncompact manifolds , 1979 .

[33]  N. Gaffke,et al.  On a class of extremal problems in statistics , 1981 .

[34]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[35]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[36]  B. Hoskins,et al.  The Mathematical Theory of Frontogenesis , 1982 .

[37]  A. Acosta.  Invariance Principles in Probability for Triangular Arrays of $B$-Valued Random Vectors and Some Applications , 1982 .

[38]  M. Gromov,et al.  A topological application of the isoperimetric inequality , 1983 .

[39]  A. Szulga.  On Minimal Metrics in the Space of Random Variables , 1983 .

[40]  Helmut Neunzert,et al.  An introduction to the nonlinear Boltzmann-Vlasov equation , 1984 .

[41]  M. Knott,et al.  On the optimal mapping of distributions , 1984 .

[42]  H. Kellerer.  Duality theorems for marginal problems , 1984 .

[43]  D. Mihalas,et al.  Foundations of Radiation Hydrodynamics , 1985 .

[44]  Ludger Rüschendorf,et al.  The Wasserstein distance and approximation theorems , 1985 .

[45]  Hitoshi Ishii,et al.  A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities , 1985 .

[46]  S. Rachev.  The Monge–Kantorovich Mass Transference Problem and Its Stochastic Applications , 1985 .

[47]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[48]  D. Stroock,et al.  Logarithmic Sobolev inequalities and stochastic Ising models , 1987 .

[49]  F. Paulin.  Topologie de Gromov équivariante, structures hyperboliques et arbres réels , 1988 .

[50]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[51]  M. Cullen,et al.  Properties of the Lagrangian Semigeostrophic Equations , 1989 .

[52]  J. A. Cuesta,et al.  Notes on the Wasserstein Metric in Hilbert Spaces , 1989 .

[53]  F. Clarke.  Methods of dynamic and nonsmooth optimization , 1989 .

[54]  S. Rachev,et al.  A characterization of random variables with minimum L 2 -distance , 1990 .

[55]  B. Maurey.  Some deviation inequalities , 1990, math/9201216.

[56]  J. Mather,et al.  Action minimizing invariant measures for positive definite Lagrangian systems , 1991 .

[57]  Michel Talagrand.  A new isoperimetric inequality for product measure and the tails of sums of independent random variables , 1991 .

[58]  Amir Dembo,et al.  Information theoretic inequalities , 1991, IEEE Trans. Inf. Theory.

[59]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[60]  P. Petersen,et al.  Manifolds near the boundary of existence , 1991 .

[61]  Jim Freeman.  Probability Metrics and the Stability of Stochastic Models , 1991 .

[62]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[63]  M. Gromov.  Sign and geometric meaning of curvature , 1991 .

[64]  G. Toscani.  New a priori estimates for the spatially homogeneous Boltzmann equation , 1992 .

[65]  J. Vázquez.  An Introduction to the Mathematical Theory of the Porous Medium Equation , 1992 .

[66]  M. Ledoux.  On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions , 1992 .

[67]  L. Saloff-Coste,et al.  A note on Poincaré, Sobolev, and Harnack inequalities , 1992 .

[68]  A. Grigor’yan.  THE HEAT EQUATION ON NONCOMPACT RIEMANNIAN MANIFOLDS , 1992 .

[69]  L. Gross.  Logarithmic Sobolev inequalities and contractivity properties of semigroups , 1993 .

[70]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[71]  J. Mather.  Variational construction of connecting orbits , 1993 .

[72]  On the Stochastic Convergence of Representations Based on Wasserstein Metrics , 1993 .

[73]  N. Trudinger.  Isoperimetric inequalities for quermassintegrals , 1994 .

[74]  M. Knott,et al.  On a generalization of cyclic monotonicity and distances among random vectors , 1994 .

[75]  A. Kasue,et al.  SPECTRAL CONVERGENCE OF RIEMANNIAN MANIFOLDS, II , 1994 .

[76]  Y. Otsu,et al.  The Riemannian structure of Alexandrov spaces , 1994 .

[77]  Wilfrid Gangbo.  An elementary proof of the polar factorization of vector-valued functions , 1994 .

[78]  Mass of rays in Alexandrov spaces of nonnegative curvature , 1994 .

[79]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[80]  R. McCann.  A convexity theory for interacting gases and equilibrium crystals , 1994 .

[81]  L. Rüschendorf,et al.  A general duality theorem for marginal problems , 1995 .

[82]  L. Rüschendorf.  Optimal solutions of multivariate coupling problems , 1995 .

[83]  P. Koskela,et al.  Sobolev meets Poincaré , 1995 .

[84]  R. McCann.  Existence and uniqueness of monotone measure-preserving maps , 1995 .

[85]  W. Gangbo,et al.  Optimal maps in Monge's mass transport problem , 1995 .

[86]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[87]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[88]  Leonid Prigozhin,et al.  Variational model of sandpile growth , 1996, European Journal of Applied Mathematics.

[89]  Piotr Hajłasz,et al.  @ 1996 Kluwer Academic Publishers. Printed in the Netherlands. Sobolev Spaces on an Arbitrary Metric Space , 1994 .

[90]  Giuseppe Toscani,et al.  The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation , 1996 .

[91]  M. Talagrand.  New concentration inequalities in product spaces , 1996 .

[92]  K. Marton.  A measure concentration inequality for contracting markov chains , 1996 .

[93]  M. Talagrand.  Transportation cost for Gaussian and other product measures , 1996 .

[94]  M. Ledoux,et al.  Isoperimetry and Gaussian analysis , 1996 .

[95]  L. Rüschendorf,et al.  Duality and perfect probability spaces , 1996 .

[96]  Jeff Cheeger,et al.  Lower bounds on Ricci curvature and the almost rigidity of warped products , 1996 .

[97]  E Weinan,et al.  Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics , 1996 .

[98]  L. Rüschendorf.  On c-optimal random variables , 1996 .

[99]  Convergence des fonctions monotones , 1996 .

[100]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[101]  L. Rüschendorf,et al.  On Optimal Multivariate Couplings , 1997 .

[102]  J. Urbas.  On the second boundary value problem for equations of Monge-Ampère type. , 1997 .

[103]  A. Dembo.  Information inequalities and concentration of measure , 1997 .

[104]  Carlos Matrán,et al.  Optimal Transportation Plans and Convergence in Distribution , 1997 .

[105]  On the Monotonicity of Optimal Transportation Plans , 1997 .

[106]  Shunhui Zhu,et al.  The Comparison Geometry of Ricci Curvature , 1997 .

[107]  R. Mañé,et al.  Lagrangian flows: The dynamics of globally minimizing orbits , 1997 .

[108]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[109]  W. Gangbo,et al.  Optimal maps for the multidimensional Monge-Kantorovich problem , 1998 .

[110]  J. Heinonen,et al.  Quasiconformal maps in metric spaces with controlled geometry , 1998 .

[111]  A. Fathi.  Sur la convergence du semi-groupe de Lax-Oleinik , 1998 .

[112]  Felix Otto,et al.  Lubrication approximation with prescribed nonzero contact anggle , 1998 .

[113]  A Stochastic Model for Growing Sandpiles and its Continuum Limit , 1998 .

[114]  Alexander Grigor'yan,et al.  Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds , 1999 .

[115]  Giuseppe Toscani,et al.  Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation , 1999 .

[116]  M. Ledoux.  Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[117]  R. McCann.  Exact solutions to the transportation problem on the line , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[118]  Dario Cordero-Erausquin.  Sur le transport de mesures périodiques , 1999 .

[119]  J. Roquejoffre,et al.  Remarks on the long time behaviour of the solutions of hamilton-jacobi equations , 1999 .

[120]  Paul-Marie Samson,et al.  Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes , 2000 .

[121]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[122]  G. Paternain,et al.  The Palais-Smale Condition and Mañé's Critical Values , 2000 .

[123]  J. Urbas.  Some interior regularity results for solutions of Hessian equations , 2000 .

[124]  Jeff Cheeger,et al.  On the structure of spaces with Ricci curvature bounded below. II , 2000 .

[125]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[126]  A. Arnold,et al.  On generalized Csiszár-Kullback inequalities , 2000 .

[127]  J. Urbas.  An interior curvature bound for hypersurfaces of prescribed $k$-th mean curvature , 2000 .

[128]  R. Jordan,et al.  Variational formulations for Vlasov–Poisson–Fokker–Planck systems , 2000 .

[129]  W. Gangbo,et al.  Shape recognition via Wasserstein distance , 2000 .

[130]  L. Rüschendorf,et al.  Numerical and analytical results for the transportation problem of Monge-Kantorovich , 2000 .

[131]  J. Heinonen.  Lectures on Analysis on Metric Spaces , 2000 .

[132]  X. Menguy.  Examples of nonpolar limit spaces , 2000 .

[133]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[134]  Florent Malrieu,et al.  Logarithmic Sobolev Inequalities for Some Nonlinear Pde's , 2001 .

[135]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[136]  Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise , 2000, nlin/0009028.

[137]  C. Villani,et al.  Comment on: “Hypercontractivity of Hamilton–Jacobi equations”, by S. Bobkov, I. Gentil and M. Ledoux , 2001 .

[138]  R. McCann,et al.  A Riemannian interpolation inequality à la Borell, Brascamp and Lieb , 2001 .

[139]  M. Sever.  An existence theorem in the large for zero-pressure gas dynamics , 2001, Differential and Integral Equations.

[140]  Diogo Aguiar Gomes,et al.  A stochastic analogue of Aubry-Mather theory , 2001 .

[141]  N. Trudinger,et al.  On the Monge mass transfer problem , 2001 .

[142]  Jean-Michel Roquejoffre,et al.  Convergence to steady states or periodic solutions in a class of Hamilton–Jacobi equations , 2001 .

[143]  S. Bobkov,et al.  Hypercontractivity of Hamilton-Jacobi equations , 2001 .

[144]  C. Villani,et al.  On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation , 2001 .

[145]  John Urbas.  THE SECOND BOUNDARY VALUE PROBLEM FOR A CLASS OF HESSIAN EQUATIONS , 2001 .

[146]  LOGARITHMIC SOBOLEV INEQUALITIES: CONDITIONS AND COUNTEREXAMPLES , 2001 .

[147]  R. McCann.  Polar factorization of maps on Riemannian manifolds , 2001 .

[148]  Some geometric properties of the Bakry-Émery-Ricci tensor , 2002, math/0211065.

[149]  R. McCann,et al.  Uniqueness and transport density in Monge's mass transportation problem , 2002 .

[150]  L. Rüschendorf,et al.  On the n-Coupling Problem , 2002 .

[151]  First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models , 2002, math/0211131.

[152]  Thierry Goudon,et al.  Fourier-Based Distances and Berry-Esseen Like Inequalities for Smooth Densities , 2002 .

[153]  Transportation cost inequalities on path spaces over Riemannian manifolds , 2002 .

[154]  Yu Ding.  Heat kernels and Green’s functions on limit spaces , 2002 .

[155]  Applications of the Monge - Kantorovich theory , 2002 .

[156]  D. Edmunds.  ASPECTS OF SOBOLEV-TYPE INEQUALITIES (London Mathematical Society Lecture Note Series 289) By LAURENT SALOFF-COSTE: 190 pp., £24.95 (LMS members' price £18.71), ISBN 0 521 00607 4 (Cambridge University Press, 2001) , 2002 .

[157]  U. Frisch,et al.  A reconstruction of the initial conditions of the Universe by optimal mass transportation , 2001, Nature.

[158]  V. V. Buldygin,et al.  Brunn-Minkowski inequality , 2000 .

[159]  Dario Cordero-Erausquin,et al.  Some Applications of Mass Transport to Gaussian-Type Inequalities , 2002 .

[160]  Luc Rey-Bellet,et al.  Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics , 2002 .

[161]  Jonathan C. Mattingly,et al.  Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , 2002 .

[162]  N. Trudinger,et al.  Hessian Measures III , 2002 .

[163]  Manuel del Pino,et al.  Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆ , 2002 .

[164]  Diogo A. Gomes,et al.  Linear programming interpretations of Mather's variational principle , 2002 .

[165]  I. Gentil.  Ultracontractive bounds on Hamilton–Jacobi solutions , 2002 .

[166]  C. Villani.  Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory , 2002 .

[167]  A. Üstünel,et al.  Measure transport on Wiener space and the Girsanov theorem , 2002 .

[168]  On stochastic domination in the Brascamp–Lieb framework , 2003, Mathematical Proceedings of the Cambridge Philosophical Society.

[169]  M. Cullen,et al.  The Fully Compressible Semi-Geostrophic System from Meteorology , 2003 .

[170]  C. Villani,et al.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , 2003 .

[171]  T. Shioya,et al.  Sobolev and Dirichlet spaces over maps between metric spaces , 2003 .

[172]  Cédric Villani,et al.  Optimal transportation, dissipative PDE’s and functional inequalities , 2003 .

[173]  K. Guittet.  On the Time-Continuous Mass Transport Problem and Its Approximation by Augmented Lagrangian Techniques , 2003, SIAM J. Numer. Anal..

[174]  Allen Tannenbaum,et al.  On the Monge-Kantorovich problem and image warping , 2003 .

[175]  Vladimir Oliker,et al.  Optical Design of Two-reflector Systems, the Monge-Kantorovich Mass Transfer Problem and Fermat's Principle , 2003 .

[176]  F. Malrieu.  Convergence to equilibrium for granular media equations and their Euler schemes , 2003 .

[177]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

[178]  Feng-Yu Wang.  Probability distance inequalities on Riemannian manifolds and path spaces , 2004 .

[179]  Talagrand’s T2-transportation Inequality w.r.t. a Uniform Metric for Diffusions , 2004 .

[180]  K. Siburg.  The Principle of Least Action in Geometry and Dynamics , 2004 .

[181]  Toshio Mikami,et al.  Duality Theorem for Stochastic Optimal Control Problem , 2006 .

[182]  Reinhard F. Werner.  The uncertainty relation for joint measurement of position and momentum , 2004, Quantum Inf. Comput..

[183]  C. Villani,et al.  A MASS-TRANSPORTATION APPROACH TO SHARP SOBOLEV AND GAGLIARDO-NIRENBERG INEQUALITIES , 2004 .

[184]  Uriel Frisch,et al.  Application of optimal transportation theory to the reconstruction of the early Universe. , 2004 .

[185]  A. Guillin,et al.  Transportation cost-information inequalities and applications to random dynamical systems and diffusions , 2004, math/0410172.

[186]  A. Guillin,et al.  Modified logarithmic Sobolev inequalities and transportation inequalities , 2004, math/0405520.

[187]  F. Hérau,et al.  Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential , 2004 .

[188]  T. Mikami.  Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes , 2004 .

[189]  T. Mikami.  A Simple Proof of Duality Theorem for Monge-Kantorovich Problem , 2004 .

[190]  Jonathan C. Mattingly,et al.  Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing , 2004, math/0406087.

[191]  Gilles Hargé.  A convex/log-concave correlation inequality for Gaussian measure and an application to abstract Wiener spaces , 2004 .

[192]  Giuseppe Toscani,et al.  Finite speed of propagation in porous media by mass transportation methods , 2004 .

[193]  Lei Zhu,et al.  Optimal Mass Transport for Registration and Warping , 2004, International Journal of Computer Vision.

[194]  Antonio Siconolfi,et al.  Existence of C1 critical subsolutions of the Hamilton-Jacobi equation , 2004 .

[195]  Luigi De Pascale,et al.  Sharp summability for Monge Transport density via Interpolation , 2004 .

[196]  C. Villani,et al.  Ricci curvature for metric-measure spaces via optimal transport , 2004, math/0412127.

[197]  Karl-Theodor Sturm,et al.  Transport inequalities, gradient estimates, entropy and Ricci curvature , 2005 .

[198]  Transportation cost inequalities on path and loop groups , 2005 .

[199]  Modified logarithmic Sobolev inequalities in null curvature , 2005, math/0503585.

[200]  Felix Otto,et al.  Eulerian Calculus for the Contraction in the Wasserstein Distance , 2005, SIAM J. Math. Anal..

[201]  Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds , 2005 .

[202]  Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation , 2004, math/0402449.

[203]  Adam M. Oberman,et al.  Computing the Effective Hamiltonian using a Variational Approach , 2004, Proceedings of the 44th IEEE Conference on Decision and Control.

[204]  Balls have the worst best Sobolev inequalities , 2005 .

[205]  N. Gozlan,et al.  A large deviation approach to some transportation cost inequalities , 2005, math/0510601.

[206]  Anatoly M. Vershik,et al.  Kantorovich Metric: Initial History and Little-Known Applications , 2005 .

[207]  J. Demange.  Porous media equation and Sobolev inequalities under negative curvature , 2005 .

[208]  Karl-Theodor Sturm,et al.  Convex functionals of probability measures and nonlinear diffusions on manifolds , 2005 .

[209]  Robert J. McCann,et al.  Fast Diffusion to Self-Similarity: Complete Spectrum, Long-Time Asymptotics, and Numerology , 2005 .

[210]  Transport optimal et courbure de Ricci , 2005 .

[211]  I. Fragalà,et al.  Continuity of an optimal transport in Monge problem , 2005 .

[212]  G. Loeper.  On the regularity of maps solutions of optimal transportation problems , 2005 .

[213]  N. Trudinger,et al.  Regularity of Potential Functions of the Optimal Transportation Problem , 2005 .

[214]  L. Kantorovich.  On a Problem of Monge , 2006 .

[215]  E. Lutwak,et al.  Optimal Sobolev norms and the Lp Minkowski problem , 2006 .

[216]  G. Loeper.  The Reconstruction Problem for the Euler-Poisson System in Cosmology , 2006 .

[217]  P. Cattiaux,et al.  Probabilistic approach for granular media equations in the non-uniformly convex case , 2006, math/0603541.

[218]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[219]  C. Villani,et al.  Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media , 2006 .

[220]  Talagrand’s T2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations* , 2006 .

[221]  G. Loeper,et al.  A Fully Nonlinear Version of the Incompressible Euler Equations: The Semigeostrophic System , 2006, SIAM J. Math. Anal..

[222]  T. Kurtz,et al.  Large Deviations for Stochastic Processes , 2006 .

[223]  Hitoshi Ishii,et al.  Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space , 2006 .

[224]  H. Ishii,et al.  Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator , 2006 .

[225]  R. McCann,et al.  Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport , 2006 .

[226]  N. Gozlan.  Integral criteria for transportation cost inequalities , 2006, math/0601384.

[227]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[228]  J. Lott.  Some Geometric Calculations on Wasserstein Space , 2006, math/0612562.

[229]  Liming Wu,et al.  Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition , 2006 .

[230]  Optimal Transport and Ricci Curvature for Metric- Measure Spaces , 2006, math/0610154.

[231]  Karl-Theodor Sturm,et al.  On the geometry of metric measure spaces , 2006 .

[232]  S. Lisini,et al.  Characterization of absolutely continuous curves in Wasserstein spaces , 2006 .

[233]  Cédric Villani,et al.  Mathematics of Granular Materials , 2006 .

[234]  Alexander Lytchak,et al.  Open map theorem for metric spaces , 2006 .

[235]  H. Ishii.  Asymptotic solutions for large time of HamiltonJacobi equations , 2006 .

[236]  Hitoshi Ishii,et al.  ASYMPTOTIC SOLUTIONS FOR LARGE-TIME OF HAMILTON-JACOBI EQUATIONS IN EUCLIDEAN $n$ SPACE(Viscosity Solution Theory of Differential Equations and its Developments) , 2007 .

[237]  Giovanni Pisante,et al.  The Semigeostrophic Equations Discretized in Reference and Dual Variables , 2007 .

[238]  M. Meckes.  Some Remarks on Transportation Cost and Related Inequalities , 2004, math/0405376.

[239]  A. Pratelli,et al.  On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation , 2007 .

[240]  Alessio Figalli.  Existence, Uniqueness, and Regularity of Optimal Transport Maps , 2007, SIAM J. Math. Anal..

[241]  On the Hessian of the optimal transport potential , 2007 .

[242]  Walter Schachermayer,et al.  Characterization of optimal transport plans for the Monge-Kantorovich problem , 2007, 0711.1268.

[243]  Ansgar Jüngel,et al.  The Derrida-Lebowitz-Speer-Spohn Equation: Existence, NonUniqueness, and Decay Rates of the Solutions , 2008, SIAM J. Math. Anal..

[244]  Patrick Cattiaux,et al.  A CRITERION FOR TALAGRAND'S QUADRATIC TRANSPORTATION COST INEQUALITY. , 2008 .

[245]  Toshio Mikami,et al.  Optimal Transportation Problem by Stochastic Optimal Control , 2008, SIAM J. Control. Optim..

[246]  J. Demange.  Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature , 2008 .

[247]  W. Desch,et al.  An elementary proof of the triangle inequality for the Wasserstein metric , 2008 .

[248]  Felix Otto,et al.  Continuity of Velocity Gradients in Suspensions of Rod–like Molecules , 2008 .

[249]  Shin-ichi Ohta,et al.  Gradient flows on Wasserstein spaces over compact Alexandrov spaces , 2009 .

[250]  Wilfrid Gangbo,et al.  Euler–Poisson Systems as Action-Minimizing Paths in the Wasserstein Space , 2009 .

[251]  K. Marton.  Measure concentration for Euclidean distance in the case of dependent random variables , 2004, math/0410168.