简介
At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book鈥檚 value as a most welcome reference text on this subject.
目录
Preface 5
Contents 10
Conventions 13
Introduction 19
1 Couplings and changes of variables 21
2 Three examples of coupling techniques 37
3 The founding fathers of optimal transport 45
Part I Qualitative description of optimal transport 54
4 Basic properties 56
5 Cyclical monotonicity and Kantorovich duality 63
6 The Wasserstein distances 105
7 Displacement interpolation 124
8 The Monge\u2013Mather shortening principle 174
9 Solution of the Monge problem I: Global approach 215
10 Solution of the Monge problem II: Local approach 224
11 The Jacobian equation 282
12 Smoothness 289
13 Qualitative picture 341
Part II Optimal transport and Riemannian geometry 360
14 Ricci curvature 363
15 Otto calculus 427
16 Displacement convexity I 440
17 Displacement convexity II 453
18 Volume control 497
19 Density control and local regularity 509
20 Infinitesimal displacement convexity 529
21 Isoperimetric-type inequalities 548
22 Concentration inequalities 569
23 Gradient flows I 631
24 Gradient flows II: Qualitative properties 695
25 Gradient flows III: Functional inequalities 720
Part III Synthetic treatment of Ricci curvature 731
26 Analytic and synthetic points of view 734
27 Convergence of metric-measure spaces 741
28 Stability of optimal transport 771
29 Weak Ricci curvature bounds I: Definition and Stability 792
30 Weak Ricci curvature bounds II: Geometric and analytic properties 844
Conclusions and open problems 899
References 909
List of short statements 951
List of figures 959
Index 961
Some notable cost functions 965
Contents 10
Conventions 13
Introduction 19
1 Couplings and changes of variables 21
2 Three examples of coupling techniques 37
3 The founding fathers of optimal transport 45
Part I Qualitative description of optimal transport 54
4 Basic properties 56
5 Cyclical monotonicity and Kantorovich duality 63
6 The Wasserstein distances 105
7 Displacement interpolation 124
8 The Monge\u2013Mather shortening principle 174
9 Solution of the Monge problem I: Global approach 215
10 Solution of the Monge problem II: Local approach 224
11 The Jacobian equation 282
12 Smoothness 289
13 Qualitative picture 341
Part II Optimal transport and Riemannian geometry 360
14 Ricci curvature 363
15 Otto calculus 427
16 Displacement convexity I 440
17 Displacement convexity II 453
18 Volume control 497
19 Density control and local regularity 509
20 Infinitesimal displacement convexity 529
21 Isoperimetric-type inequalities 548
22 Concentration inequalities 569
23 Gradient flows I 631
24 Gradient flows II: Qualitative properties 695
25 Gradient flows III: Functional inequalities 720
Part III Synthetic treatment of Ricci curvature 731
26 Analytic and synthetic points of view 734
27 Convergence of metric-measure spaces 741
28 Stability of optimal transport 771
29 Weak Ricci curvature bounds I: Definition and Stability 792
30 Weak Ricci curvature bounds II: Geometric and analytic properties 844
Conclusions and open problems 899
References 909
List of short statements 951
List of figures 959
Index 961
Some notable cost functions 965
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