Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[3]  L. Lecam,et al.  Convergence in distribution of stochastic processes , 1957 .

[4]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[5]  Michel Loève,et al.  Probability Theory I , 1977 .

[6]  P. Hartman.  Ordinary Differential Equations , 1965 .

[7]  J. Serrin,et al.  Sublinear functions of measures and variational integrals , 1964 .

[8]  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 .

[9]  H. Brezis.  Propriétés Régularisantes de Certains Semi-Groupes Non Linéaires , 1971 .

[10]  Michael G. Crandall,et al.  GENERATION OF SEMI-GROUPS OF NONLINEAR TRANSFORMATIONS ON GENERAL BANACH SPACES, , 1971 .

[11]  L. Young,et al.  Lectures on the Calculus of Variations and Optimal Control Theory. , 1971 .

[12]  L. Schwartz.  Radon measures on arbitrary topological spaces and cylindrical measures , 1973 .

[13]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[14]  J. Lasry,et al.  Int'egrandes normales et mesures param'etr'ees en calcul des variations , 1973 .

[15]  H. Brezis.  Interpolation classes for monotone operators , 1975 .

[16]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[17]  P. Meyer,et al.  Probabilities and potential C , 1978 .

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

[19]  M. Pierre.  Uniqueness of the solutions of ut−Δϕ(u) = 0 with initial datum a measure☆ , 1982 .

[20]  Erik J. Balder,et al.  A General Approach to Lower Semicontinuity and Lower Closure in Optimal Control Theory , 1984 .

[21]  Michael G. Crandall.  Nonlinear Semigroups and Evolution Governed by Accretive Operators. , 1984 .

[22]  Marco Degiovanni,et al.  Evolution equations with lack of convexity , 1985 .

[23]  H. Dietrich.  Zur c-konvexität und c-subdifferenzierbarkelt von funktionalen , 1988 .

[24]  Claudio Baiocchi,et al.  Discretization of Evolution Variational Inequalities , 1989 .

[25]  Mario Tosques,et al.  Curves of maximal slope and parabolic variational inequalities on non-convex constraints , 1989 .

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

[27]  S. Rachev,et al.  New duality theorems for marginal problems with some applications in stochastics , 1989 .

[28]  Luigi Ambrosio,et al.  Metric space valued functions of bounded variation , 1990 .

[29]  Stephan Luckhaus,et al.  Solutions for the two-phase Stefan problem with the Gibbs–Thomson Law for the melting temperature , 1990, European Journal of Applied Mathematics.

[30]  A. Visintin,et al.  On A Class Of Doubly Nonlinear Evolution Equations , 1990 .

[31]  M. Giaquinta,et al.  Area and the area formula , 1992 .

[32]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[33]  Pierluigi Colli,et al.  On some doubly nonlinear evolution equations in Banach spaces , 1992 .

[34]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[35]  F. Almgren,et al.  Curvature-driven flows: a variational approach , 1993 .

[36]  D. Stroock,et al.  Probability Theory: An Analytic View , 1995, The Mathematical Gazette.

[37]  Pavel Bleher,et al.  Existence and positivity of solutions of a fourth‐order nonlinear PDE describing interface fluctuations , 1994 .

[38]  Giuseppe Savaré,et al.  SINGULAR PERTURBATION AND INTERPOLATION , 1994 .

[39]  S. Luckhaus,et al.  Implicit time discretization for the mean curvature flow equation , 1995 .

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

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

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

[43]  J. Rulla,et al.  Error analysis for implicit approximations to solutions to Cauchy problems , 1996 .

[44]  Felix Otto,et al.  Doubly Degenerate Diffusion Equations as Steepest Descent , 1996 .

[45]  A. Visintin.  Models of Phase Transitions , 1996 .

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

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

[48]  Jürgen Jost,et al.  Nonpositive Curvature: Geometric And Analytic Aspects , 1997 .

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

[50]  Uwe F. Mayer,et al.  Gradient flows on nonpositively curved metric spaces and harmonic maps , 1998 .

[51]  S. Rachev,et al.  Mass transportation problems , 1998 .

[52]  D. Kinderlehrer,et al.  Approximation of Parabolic Equations Using the Wasserstein Metric , 1999 .

[53]  W. Gangbo.  THE MONGE MASS TRANSFER PROBLEM AND ITS APPLICATIONS , 1999 .

[54]  L. Evans,et al.  Differential equations methods for the Monge-Kantorovich mass transfer problem , 1999 .

[55]  Jeff Cheeger,et al.  Differentiability of Lipschitz Functions on Metric Measure Spaces , 1999 .

[56]  T. Mikami.  Dynamical Systems in the Variational Formulation of the Fokker—Planck Equation by the Wasserstein Metric , 1999 .

[57]  M. Csörnyei.  Aronszajn null and Gaussian null sets coincide , 1999 .

[58]  L. Ambrosio,et al.  A geometrical approach to monotone functions in $\mathbb R^n$ , 1999 .

[59]  Luigi Ambrosio,et al.  Rectifiable sets in metric and Banach spaces , 2000 .

[60]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[61]  Giuseppe Savare',et al.  A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations † , 2000 .

[62]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

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

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

[65]  Ansgar Jüngel,et al.  Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems , 2000, SIAM J. Math. Anal..

[66]  Giuseppe Toscani,et al.  ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS , 2001 .

[67]  R. McCann,et al.  Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs , 2001 .

[68]  Lorenzo Giacomelli,et al.  Variatonal formulation for the lubrication approximation of the Hele-Shaw flow , 2001 .

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

[70]  Ronald F. Gariepy.  FUNCTIONS OF BOUNDED VARIATION AND FREE DISCONTINUITY PROBLEMS (Oxford Mathematical Monographs) , 2001 .

[71]  T. O’Neil.  Geometric Measure Theory , 2002 .

[72]  J. Carrillo,et al.  On the Long-Time Behavior of the Quantum Fokker-Planck Equation , 2002, math-ph/0204032.

[73]  Manuel del Pino,et al.  Nonlinear diffusions, hypercontractivity and the optimal LP-Euclidean logarithmic Sobolev inequality , 2004 .

[74]  A. Mielke,et al.  A Variational Formulation of¶Rate-Independent Phase Transformations¶Using an Extremum Principle , 2002 .

[75]  Giuseppe Da Prato,et al.  Second Order Partial Differential Equations in Hilbert Spaces: Bibliography , 2002 .

[76]  J. Vázquez,et al.  Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities , 2002 .

[77]  C. Villani,et al.  Homogeneous Cooling States¶are not always¶Good Approximations to Granular Flows , 2002 .

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

[79]  M. Agueh.  Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. , 2002, math/0309410.

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

[81]  W. Gangbo,et al.  Constrained steepest descent in the 2-Wasserstein metric , 2003, math/0312063.

[82]  L. Ambrosio,et al.  Existence and stability results in the L 1 theory of optimal transportation , 2003 .

[83]  C. Villani.  Topics in Optimal Transportation , 2003 .

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

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

[86]  Karl B Glasner,et al.  A diffuse interface approach to Hele-Shaw flow , 2003 .

[87]  M. Agueh,et al.  The optimal evolution of the free energy of interacting gases and its applications , 2003 .

[88]  Luigi Ambrosio,et al.  Existence of optimal transport maps for crystalline norms , 2004 .

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

[90]  G. Prato,et al.  Elliptic operators with unbounded drift coefficients and Neumann boundary condition , 2004 .

[91]  María J. Cáceres,et al.  Long-time behavior for a nonlinear fourth-order parabolic equation , 2004 .

[92]  V. Bogachev,et al.  Existence of Solutions To Weak Parabolic Equations For Measures , 2004 .

[93]  D. Kinderlehrer,et al.  Remarks about the Flashing Rachet , 2004 .

[94]  A. Kolesnikov.  Convexity inequalities and optimal transport of infinite-dimensional measures , 2004 .

[95]  A. Üstünel,et al.  Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space , 2004 .

[96]  L. C. Evans,et al.  Diffeomorphisms and Nonlinear Heat Flows , 2005, SIAM Journal on Mathematical Analysis.

[97]  Anton Arnold,et al.  Refined convex Sobolev inequalities , 2005 .

[98]  Giuseppe Savaré,et al.  NONLINEAR EVOLUTION GOVERNED BY ACCRETIVE OPERATORS IN BANACH SPACES:: ERROR CONTROL AND APPLICATIONS , 2006 .

[99]  Giuseppe Savaré,et al.  Gradient flows of non convex functionals in Hilbert spaces and applications , 2006 .

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

[101]  Giovanni Alberti,et al.  A geometrical approach to monotone functions in R n , 2007 .

[102]  O. Gaans.  Probability measures on metric spaces , 2022 .