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Random Trees, Lévy Processes and Spatial Branching Processes
Thomas Duquesne, Université de Paris-Sud, Orsay, France, and Jean-François Le Gall, École Normale Supérieure, Paris, France
A publication of the Société Mathématique de France.
Astérisque
2002; 146 pp; softcover
Number: 281
ISBN-10: 2-85629-128-7
ISBN-13: 978-2-85629-128-3
List Price: US$33
Individual Members: US$29.70
Order Code: AST/281
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In the book, the authors investigate the genealogical structure of general critical or subcritical continuous-state branching processes. Analogously to the coding of a discrete tree by its contour function, this genealogical structure is coded by a real-valued stochastic process called the height process, which is itself constructed as a local time functional of a Lévy process with no negative jumps. They present a detailed study of the height process and of an associated measure-valued process called the exploration process, which plays a key role in most applications. Under suitable assumptions, it is proven that whenever a sequence of rescaled Galton-Watson processes converges in distribution, their genealogies also converge to the continuous branching structure coded by the appropriate height process. The authors apply this invariance principle to various asymptotics for Galton-Watson trees and then use the duality properties of the exploration process to compute explicitly the distribution of the reduced tree associated with Poissonnian marks in the height process and the finite-dimensional marginals of the so-called stable continuous tree. This last calculation generalizes to the stable case a result of Aldous for the Brownian continuum random tree. Finally, they combine the genealogical structure with an independent spatial motion to develop a new approach to superprocesses with a general branching mechanism. In this setting, they derive certain explicit distributions, such as the law of the spatial reduced tree in a domain, consisting of the collection of all historical paths that hit the boundary.

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.

Readership

Graduate students and research mathematicians interested in probability theory and stochastic processes.

Table of Contents

  • Introduction
  • The height process
  • Convergence of Galton-Watson trees
  • Marginals of continuous trees
  • The Lévy snake
  • Bibliography
  • Notation index
  • Index
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