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Milieux aléatoires
Nina Gantert, Universität Karlsruhe, Germany, Josselin Garnier, Université Paul Sabatier, Toulouse, France, Stefano Olla, Université de Paris 9-Dauphine, France, Zhan Shi, Université de Paris VI, France, and Alain-Sol Sznitman, Mathematik, ETH-Zentrum, Zürich, Switzerland
A publication of the Société Mathématique de France.
Panoramas et Synthèses
2002; 159 pp; softcover
Number: 12
ISBN-10: 2-85629-127-9
ISBN-13: 978-2-85629-127-6
List Price: US$33
Member Price: US$26.40
Order Code: PASY/12
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Random media are natural models for nonhomogeneous materials which possess some kind of statistical regularity. The study of stochastic processes in random media is currently an active field of research, and new techniques have recently been developed, including mathematical forms of renormalization. Those techniques apply to models which are much more delicate than exactly soluble ones or even reversible ones.

The session, "États de la Recherche", presented the state of the art in the field and brought it to a large portion of the scientific community. Based on the notes of the courses delivered during the session, this volume is composed of five articles and a general introduction, where all basic notions from probability theory are defined. The introduction and the style of the articles make the volume readable by nonspecialists.

The article by Alain Sznitman studies the survival of Brownian motion moving among randomly located obstacles, and the ballistic behavior of the random walk in random media on the \(d\ge 2\)-dimensional lattice. This illustrates the role of atypical pockets in the medium and of abnormally small eigenvalues. The second article, by Zhan Shi, presents the analysis via stochastic calculus of Sinaï's random walk and of the one dimensional diffusion in a Brownian potential. Nina Gantert studies the random walk on a random Galton-Watson tree, in particular, the probability of rare events. Stefano Olla studies random homogenization, taking the point of view of the environment seen from the particle, as well as applications to interacting particle systems. In the last article, Josselin Garnier studies wave propagation in random media, the competition between nonlinear and random effects, and solitons in this framework.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians.

Table of Contents

  • Introduction générale
  • A.-S. Sznitman -- Milieux aléatoires et petites valeurs propres
  • N. Gantert -- Galton-Watson trees as random environments
  • Z. Shi -- Sinai's walk via stochastic calculus
  • S. Olla -- Central limit theorems for tagged particles and for diffusions in random environment
  • J. Garnier -- Wave propagation in one-dimensional random media
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