最优输运理论专题(第2版)(英文版)(精)/美国数学会经典影印系列

1781年，Gaspard Monge定义了“最优输运”问 题（即以可能的最小工作量进行质量转移），并将其 应用于工程。1942年，Leonid Kantorovich将新生 的线性规划用于Monge问题，并将其应用于经济。 1987年，Yann Brenier利用最优输运证明了一个保持 映射的度量集上新的规划定理，并将其应用于流体力 学。 每一个这样的贡献都标志着一个完整数学理论的 开端，它有很多意料不到的分支。当前，研究人员从 极其多样化的视角来使用和研究Monge-Kantorovich 问题，这包括概率论、泛函分析、等周问题、偏微分 方程乃至气象学。 塞德里克·维拉尼著的这本《最优输运理论专题 （第2版）（英文版）》源于一门研究生课，可用作 最优输运领域的入门书，概述了最近15年该领域的研 究全貌。本书面向研究生和科研人员，理论和应用并 重，读者只需熟悉测度论和泛函分析的基础知识。

Preface of the Second Edition Preface of the First Edition Notation Introduction §1. Formulation of the optimal transportation problem §2. Basic questions §3. Overview of the course Chapter 1. The Kantorovich Duality §1.1. General duality §1.2. Distance cost functions §1.3. Appendix: A duality argument in Cb(X § Y) §1.4. Appendix: {0, 1}-valued costs and Strassen's theorem Chapter 2. Geometry of Optimal Transportation §2.1. A duality-based proof for the quadratic cost §2.2. The real line §2.3. Alternative arguments §2.4. Generalizations to other costs §2.5. More on c-concave functions Chapter 3. Brenier's Polar Factorization Theorem §3.1. Rearrangements and polar factorization §3.2. Historical motivations: fluid mechanics §3.3. Proof of Brenier's polar factorization theorem §3.4. Related facts Chapter 4. The Monge-Ampere Equation §4.1. Informal presentation §4.2. Regularity §4.3. Open problems Chapter 5. Displacement Interpolation and Displacement Convexity §5.1. Displacement interpolation §5.2. Displacement convexity §5.3. Application: uniqueness of ground state §5.4. The Eulerian point of view Chapter 6. Geometric and Gaussian Inequalities §6.1. Brunn-Minkowski and Prekopa-Leindler inequalities §6.2. The Alesker-Dar-Milman diffeomorphism §6.3. Gaussian inequalities §6.4. Sobolev inequalities Chapter 7. The Metric Side of Optimal Transportation §7.1. Monge-Kantorovich distances §7.2. Topological properties §7.3. The real line §7.4. Behavior under rescaled convolution §7.5. An application to the Boltzmann equation §7.6. Linearization Chapter 8. A Differential Point of View on Optimal Transportation §8.1. A differential formulation of optimal transportation §8.2. Differential calculus in (P(Rn), W2) §8.3. Monge-Kantorovich induced dynamics §8.4. Time-discretization §8.5. Differentiability of the quadratic Wasserstein distance §8.6. Non-quadratic costs Chapter 9. Entropy Production and Transportation Inequalities §9.1. More on optimal-transportation induced dissipative equations §9.2. Logarithmic Sobolev inequalities §9.3. Talagrand inequalities §9.4. HWI inequalities §9.5. Nonlinear generalizations: internal energy §9.6. Nonlinear generalizations: interaction energy Chapter 10. Problems List of Problems Bibliography Table of Short Statements Index