Contemporary Mathematics Volume 409, 2006 Steepest descent flows and applications to spaces of probability measures Luigi Ambrosio 1. Introduction These notes are the outcome of a series of seminars held in Santander (Spain) in July 2004, concerning some of the results contained in the book [AGS04]. The first part of these lecture notes is devoted to the study of gradient flows in a general metric setting, while the second part is concerned with the metric space of proba- bility measures endowed with the Wasserstein distance. In the first part (section 2) I start by recalling the main concepts of the classical theory of gradient flows in Hilbert spaces. Then, I introduce the definition of steepest descent flow, which gen- eralizes the idea of gradient flow to a purely metric setting. To this aim, I make use of the concepts of metric derivative and local slope (see also [DGMT80, Amb95]). Then, I deduce some useful properties typically satisfied by functionals which are convex along geodesics. Such properties also involve the notion of upper gradient (see [HK98]). These properties ensure the convergence of the implicit Euler time discretization scheme. I also establish the link between our metric framework and the well known results of the classical theory on Hilbert spaces. Finally, I recall the conditions needed to have uniqueness and error estimates of the approximating scheme. In the second part (section 3) I study the differentiable structure of the Wasser- stein space to give an equivalent concept of gradient flow. The Wasserstein distance arises as a useful tool to study the asymptotic behaviour of solutions to certain PDEs enjoying a gradient flow structure with respect to it. Such gradient flow structure is closely related to some logarithmic Sobolev inequalities, which ensure convergence towards the asymptotic states. This point of view has been widely de- veloped in the recent years (see [Tos96, Tos97, JK098, CTOO, OttOl, OVOO, DPD02, Agu02, Agu03, CMV] among the others.). The Wasserstein distance could seem very abstract, but in fact it is linked to a very intuitive problem: the mass transportation problem which can be formulated in terms of a pile of sand and a hole. The problem is how to transport the pile into the hole with the least Notes taken and further elaborated by Marfa J. Caceres (Departamento de Matematica Apli- cada, Universidad de Granada, 18071 Granada, Spain, email: and Marco Di Francesco (Sezione di Matematica per l'Ingegneria, Universita dell' Aquila, Piazzale Pontieri, Mon- teluco di Roio, 167100 L'Aquila (Italy), email: @2006 American Mathematical Society
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