Author	Ambrosio, Luigi. author
Title	Gradient Flows [electronic resource] : in Metric Spaces and in the Space of Probability Measures / by Luigi Ambrosio, Nicola Gigli, Giuseppe Savarรฉ
Imprint	Basel : Birkhรคuser Basel, 2008
Edition	Second Edition
Connect to	http://dx.doi.org/10.1007/978-3-7643-8722-8
Descript	IX, 334 p. online resource

In this edition we made minor corrections kindly pointed out to us by some c- leagues, and we updated and expanded the bibliography. We have not included the developments of the theory of gradient ?ows occurred in the last three years, asgradient?ows inspaceswith Alexandrovcurvaturebounds [135] (see also[119]) and Fokker-Planck equations in in?nite-dimensional spaces [18], largely based on the ideas developed in the book. We also mention the long survey paper [17], more focussed on gradient ?ows in Euclidean spaces with respect to the quadratic Wasserstein distance, where the notion of Evolution Variational Inequality is d- cussed more in detail, and the monumental book of C. Villani [147], which will surely become the standard reference for the theory of Optimal Transport and its applications to geometry and PDE's. Pisa and Pavia, January 2008 Introduction This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 devoted to gradient ?ows in the L -Wasserstein space of probability measures on 2 a separableHilbert space X endowedwith the WassersteinL metric (we consider p the L -Wasserstein distance, p? (1,?), as well)

Notation -- Notation -- Gradient Flow in Metric Spaces -- Curves and Gradients in Metric Spaces -- Existence of Curves of Maximal Slope and their Variational Approximation -- Proofs of the Convergence Theorems -- Uniqueness, Generation of Contraction Semigroups, Error Estimates -- Gradient Flow in the Space of Probability Measures -- Preliminary Results on Measure Theory -- The Optimal Transportation Problem -- The Wasserstein Distance and its Behaviour along Geodesics -- Absolutely Continuous Curves in p(X) and the Continuity Equation -- Convex Functionals in p(X) -- Metric Slope and Subdifferential Calculus in (X) -- Gradient Flows and Curves of Maximal Slope in p(X)

1. Mathematics
2. Global differential geometry
3. Distribution (Probability theory)
4. Mathematics
5. Measure and Integration
6. Differential Geometry
7. Probability Theory and Stochastic Processes