{Euclidean, metric, and Wasserstein} gradient flows: an overview

• Published in: Bulletin of mathematical sciences Vol. 7; no. 1; pp. 87 - 154
• Main Author: Santambrogio, Filippo
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2017; World Scientific Publishing Co. Pte., Ltd
• Subjects: Mathematics; Mathematics and Statistics; Fokker–Planck equation; Subdifferential; Numerical methods; Heat flow; 35K05; 54E35; Optimal transport; Contractivity; Metric measure spaces; 34G25; Cauchy problem; Analysis in metric spaces; 49J45; 00-02; 49M29; Wasserstein distances; 49Q20
• ISSN: 1664-3607; 1664-3615
• DOI: 10.1007/s13373-017-0101-1

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan–Kinderlehrer–Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.