最优输运 第2分册 英文PDF电子书下载

Introduction 1

1 Couplings and changes of variables 5

2 Three examples of coupiing 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 Ⅰ 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