Optimal Transport: Old and New (Grundlehren der mathematischen Wissenschaften Book 338)

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’s value as a most welcome reference text on this subject. Front Matter....Pages I-XXII Front Matter....Pages 1-3 Couplings and changes of variables....Pages 5-20 Three examples of coupling techniques....Pages 21-28 The founding fathers of optimal transport....Pages 29-37 Front Matter....Pages 39-41 Basic properties....Pages 43-49 Cyclical monotonicity and Kantorovich duality....Pages 51-92 The Wasserstein distances....Pages 93-111 Displacement interpolation....Pages 113-162 The Monge—Mather shortening principle....Pages 163-203 Solution of the Monge problem I: global approach....Pages 205-213 Solution of the Monge problem II: Local approach....Pages 215-272 The Jacobian equation....Pages 273-279 Smoothness....Pages 281-332 Qualitative picture....Pages 333-351 Front Matter....Pages 353-356 Ricci curvature....Pages 357-420 Otto calculus....Pages 421-433 Displacement convexity I....Pages 435-447 Displacement convexity II....Pages 449-492 Volume control....Pages 493-504 Density control and local regularity....Pages 505-524 Infinitesimal displacement convexity....Pages 525-543 Front Matter....Pages 353-356 Isoperimetric-type inequalities....Pages 545-565 Concentration inequalities....Pages 567-628 Gradient flows I....Pages 629-692 Gradient flows II: Qualitative properties....Pages 693-717 Gradient flows III: Functional inequalities....Pages 719-729 Front Matter....Pages 731-734 Analytic and synthetic points of view....Pages 735-741 Convergence of metric-measure spaces....Pages 743-772 Stability of optimal transport....Pages 773-793 Weak Ricci curvature bounds I: Definition and Stability....Pages 795-846 Weak Ricci curvature bounds II: Geometric and analytic properties....Pages 847-901 Back Matter....Pages 904-976