Optimal Mass Transport on Euclidean Spaces

A pedagogical introduction to the key ideas and theoretical foundation of optimal mass transport for a graduate course or self-study.

Francesco Maggi (Author)

9781009179706, Cambridge University Press

Hardback, published 16 November 2023

316 pages
23.5 x 15.9 x 2.5 cm, 0.63 kg

'The author brings original and pedagogical ideas and illuminating remarks to his presentation, so his book is absolutely worth working with for teaching, solo learning, reading groups and research … Francesco Maggi's book can be recommended to anybody interested in the topic.' Nicolas Juillet, MathSciNet

Optimal mass transport has emerged in the past three decades as an active field with wide-ranging connections to the calculus of variations, PDEs, and geometric analysis. This graduate-level introduction covers the field's theoretical foundation and key ideas in applications. By focusing on optimal mass transport problems in a Euclidean setting, the book is able to introduce concepts in a gradual, accessible way with minimal prerequisites, while remaining technically and conceptually complete. Working in a familiar context will help readers build geometric intuition quickly and give them a strong foundation in the subject. This book explores the relation between the Monge and Kantorovich transport problems, solving the former for both the linear transport cost (which is important in geometric applications) and for the quadratic transport cost (which is central in PDE applications), starting from the solution of the latter for arbitrary transport costs.

Preface
Notation
Part I. The Kantorovich Problem: 1. An introduction to the Monge problem
2. Discrete transport problems
3. The Kantorovich problem
Part II. Solution of the Monge Problem with Quadratic Cost: the Brenier-McCann Theorem: 4. The Brenier theorem
5. First order differentiability of convex functions
6. The Brenier-McCann theorem
7. Second order differentiability of convex functions
8. The Monge-Ampère equation for Brenier maps
Part III. Applications to PDE and the Calculus of Variations and the Wasserstein Space: 9. Isoperimetric and Sobolev inequalities in sharp form
10. Displacement convexity and equilibrium of gases
11. The Wasserstein distance W2 on P2(Rn)
12. Gradient flows and the minimizing movements scheme
13. The Fokker-Planck equation in the Wasserstein space
14. The Euler equations and isochoric projections
15. Action minimization, Eulerian velocities and Otto's calculus
Part IV. Solution of the Monge Problem with Linear Cost: the Sudakov Theorem: 16. Optimal transport maps on the real line
17. Disintegration
18. Solution to the Monge problem with linear cost
19. An introduction to the needle decomposition method
Appendix A: Radon measures on Rn and related topics
Appendix B: Bibliographical Notes
Bibliography
Index.

Subject Areas: Calculus & mathematical analysis [PBK]