Brownian Motion, Obstacles and Random Media

ISBN-13: 9783642084201 / Angielski / Miękka / 2011 / 357 str.

Alain-Sol Sznitman

Opis

The principal purpose of this book is to provide an account of the circle of ideas, results and techniques, which emerged roughly over the last ten years in the study of Brownian motion and random obstacles. The accumulation of results in many separate sources eventually made it impractical, if not impossible, for the nonspecialist to gain access to the developments of the subject. This book is an attempt to remedy this situation. Part of the thrill of the investigation of Brownian motion and random obsta- cles certainly stems from its many connections with various areas of math- ematics, but also from the formal and mysterious physical heuristics which relate to it. In particular the loose concept of pockets of low local eigenval- ues plays an important role in the study of Brownian motion and random obstacles, and also represents a paradigm which has natural resonances with several other areas of random media. This last feature has increasingly be- come clear over the last few years.

Spis treści

1. The Feynman-Kac Formula and Semigroups.- 2. Some Potential Theory.- 3. Some Principal Eigenvalue Estimates.- 4. The Method of Enlargement of Obstacles.- 5. Lyapunov Exponents.- 6. Quenched Path Measure and Pinning Effect.- 7. Overview, further Results and Problems.- References.

Blurb

This book is aimed at graduate students and researchers. It provides an account for the non-specialist of the circle of ideas, results and techniques, which grew out in the study of Brownian motion and random obstacles. This subject has a rich phenomenology which exhibits certain paradigms, emblematic of the theory of random media. It also brings into play diverse mathematical techniques such as stochastic processes, functional analysis, potential theory, first passage percolation. In a first part, the book presents, in a concrete manner, background material related to the Feynman-Kac formula, potential theory, and eigenvalue estimates. In a second part, it discusses recent developments including the method of enlargement of obstacles, Lyapunov coefficients, and the pinning effect. The book also includes an overview of known results and connections with other areas of random media.