Villani C. Optimal transport: old and new. - Berlin: Springer, 2009. - 973 p. - (Grundlehren der mathematischen Wissenschaften; 338). - ISBN 978-3-540-71049-3  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

Preface ....................................................... VII Conventions .................................................. XVII Introduction .................................................... 1 1. Couplings and changes of variables .......................... 5 2. Three examples of coupling techniques ...................... 21 3. The founding fathers of optimal transport .................. 29 Part I. Qualitative description of optimal transport ......... 39 4. Basic properties ........................................... 43 5. Cyclical monotonicity and Kantorovich duality .............. 51 6. The Wasserstein distances .................................. 93 7. Displacement interpolation ................................ 113 8. The Monge—Mather shortening principle ..................... 163 9. Solution of the Monge problem I: Global approach .......... 205 10. Solution of the Monge problem II: Local approach .......... 215 11. The Jacobian equation ..................................... 273 12. Smoothness ................................................ 281 13. Qualitative picture ....................................... 333 Part II. Optimal transport and Riemannian geometry ........... 353 14. Ricci curvature ........................................... 357 15. Otto calculus ............................................. 421 16. Displacement convexity I .................................. 435 17. Displacement convexity II ................................. 449 18. Volume control ............................................ 493 19. Density control and local regularity ...................... 505 20. Infinitesimal displacement convexity ...................... 525 21. Isoperimetric-type inequalities ........................... 545 22. Concentration inequalities ................................ 567 23. Gradient flows 1 .......................................... 629 24. Gradient flows II: Qualitative properties ................. 693 25. Gradient flows III: Functional inequalities ............... 719 Part III. Synthetic treatment of Ricci curvature .............. 731 26. Analytic and synthetic points of view ..................... 735 27 Convergence of metric-measure spaces ...................... 743 28. Stability of optimal transport ............................ 773 29. Weak Ricci curvature bounds I: Definition and Stability ................................................. 795 30. Weak Ricci curvature bounds II: Geometric and analytic properties ....................................... 847 Conclusions and open problems ................................. 903 References .................................................... 915 List of short statements ...................................... 957 List of figures ............................................... 965 Index ......................................................... 967 Some notable cost functions ................................... 971