An invitation to optimal transport, Wasserstein distances, and gradient flows
This graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices. Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry. Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs. Chapter 3 utilizes the \([1,\infty)\ni p\)-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations. Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures. Chapter 5 suggests several applied topics of the optimal transport theory. Appendix A includes a set of some interesting exercises and their solutions. Appendix B outlines a proof of the disintegration theorem.
scientific article; zbMATH DE number 7375481