{ Euclidean, Metric, and Wasserstein } Gradient Flows: an overview

TL;DR

This paper provides a comprehensive overview of gradient flows in Euclidean, metric, and Wasserstein spaces, including theory, numerical methods, and recent advances in heat flow in metric measure spaces.

Contribution

It synthesizes existing theories of gradient flows across different spaces and introduces new theoretical developments in heat flow within metric measure spaces.

Findings

Convergence proof of the Jordan-Kinderleher-Otto scheme for linear Fokker-Planck equations

Introduction of optimal transport tools and geodesic convexity concepts

New theoretical insights into heat flow in metric measure spaces

Abstract

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 Savar{\'e}. 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 desciption of the Jordan-Kinderleher-Otto scheme, with proof of convergence in the easiest case: the linear Fokker-Planck equation. A discussion of other 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, Savar{\'e}, Kuwada and Ohta: the study of the heat flow in metric measure spaces.