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 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.
Highlights
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
Steepest descent curves, are a very classical topic in evolution equations: take a functional F defined on a vector space X, and, instead of looking at points x minizing F (which is related to the statical equation ∇F(x) = 0), we look, given an initial point x0, for a curve starting at x0 and trying to minimize F as fast as possible (in this case, we will solve equations of the form x′(t) = −∇F(x(t)))
The theory is particularized in the second half of [6] to the case of the metric space of probability measures endowed with the so-called Wasserstein distance coming from optimal transport, whose differential structure is widely studied in the book
Summary
The theory is particularized in the second half of [6] to the case of the metric space of probability measures endowed with the so-called Wasserstein distance coming from optimal transport, whose differential structure is widely studied in the book In this framework, the geodesic convexity results that McCann obtained in [69] are crucial to make a bridge from the general to the particular theory. It is interesting to observe that, the heat equation turns out to be a gradient flow in two different senses: it is the gradient flow of the Dirichlet energy in the L2 space, and of the entropylog(̺) in the Wasserstein space Both frameworks can be adapted from the particular case of probabilities on a domain Ω ⊂ Rd to the more general case of metric measure spaces, and the question whether the two flows coincide, or under which assumptions they do, is natural. The reader who is mainly interested gradient flows in the Wasserstein space and in PDE applications can decide to skip sections 3, 4.4 and 5, which deal on the contrary with key objects for the - very lively at the moment - subject of analysis on metric measure spaces
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