Author(s): Cédric Villani
Publisher: A.M.S.
Year: 2003
Language: english
Commentary: much better than the preceding pdf
Pages: 388
Cover......Page 1
Title page......Page 2
Contents......Page 7
Preface......Page 11
Notation......Page 15
1. Formulation of the optimal transportation problem......Page 19
2. Basic questions......Page 24
3. Overview of the course......Page 28
1.1. General duality......Page 35
1.2. Distance cost functions......Page 52
1.3. Appendix: A duality argument in Cb(XxY)......Page 57
1.4. Appendix: {0,1}-valued costs and Strassen's theorem......Page 62
Chapter 2. Geometry of Optimal Transportation......Page 65
2.1. A duality-based proof for the quadratic cost......Page 66
2.2. The real line......Page 91
2.3. Alternative arguments......Page 96
2.4. Generalizations to other costs......Page 103
2.5. More on c-concave functions......Page 121
3.1. Rearrangements and polar factorization......Page 125
3.2. Historical motivations: fluid mechanics......Page 129
3.3. Proof of Brenier' s polar factorization theorem......Page 137
3.4. Related facts......Page 140
4.1. Informal presentation......Page 143
4.2. Regularity......Page 149
4.3. Open problems......Page 159
5.1. Displacement interpolation......Page 161
5.2. Displacement convexity......Page 168
5.3. Application: uniqueness of ground state......Page 181
5.4. The Eulerian point of view......Page 183
Chapter 6. Geometric and Gaussian Inequalities......Page 201
6.1. Brunn-Minkowski and Prekopa-Leindler inequalities......Page 202
6.2. The Alesker-Dar-Milman diffeomorphism......Page 208
6.3. Gaussian inequalities......Page 210
6.4. Sobolev inequalities......Page 218
Chapter 7. The Metric Side of Optimal Transportation......Page 223
7.1. Monge-Kantorovich distances......Page 225
7.2. Topological properties......Page 230
7.3. The real line......Page 236
7.4. Behavior under resealed convolution......Page 238
7.5. An application to the Boltzmann equation......Page 241
7.6. Linearization......Page 251
Chapter 8. A Differential Point of View on Optimal Transportation......Page 255
8.1. A differential formulation of optimal transportation......Page 256
8.2. Differential calculus......Page 268
8.3. Monge-Kantorovich induced dynamics......Page 269
8.4. Time-discretization......Page 274
8.5. Differentiability of the quadratic Wasserstein distance......Page 280
8.6. Non-quadratic costs......Page 284
Chapter 9. Entropy Production and Transportation Inequalities......Page 285
9.1. More on optimal-transportation induced dissipative equations......Page 286
9.2. Logarithmic Sobolev inequalities......Page 297
9.3. Talagrand inequalities......Page 309
9.4. HWI inequalities......Page 315
9.5. Nonlinear generalizations: internal energy......Page 319
9.6. Nonlinear generalizations: interaction energy......Page 322
Chapter 10. Problems......Page 325
List of Problems......Page 326
Bibliography......Page 367
Table of Short Statements......Page 381
Index......Page 385
