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Spatial Branching Processes, Random Snakes and Partial Differential Equations

  • Book
  • © 1999

Overview

Authors:
  1. Jean-François Gall

    1. Département de Mathématiques, Ecole Normale Supérieure, Paris, France

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Part of the book series: Lectures in Mathematics. ETH Zürich (LM)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-ix
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  2. An Overview

    • Jean-François Le Gall
    Pages 1-20

  3. Continuous-state Branching Processes and Superprocesses

    • Jean-François Le Gall
    Pages 21-40

  4. The Genealogy of Brownian Excursions

    • Jean-François Le Gall
    Pages 41-51

  5. The Brownian Snake and Quadratic Superprocesses

    • Jean-François Le Gall
    Pages 53-74

  6. Exit Measures and the Nonlinear Dirichlet Problem

    • Jean-François Le Gall
    Pages 75-88

  7. Polar Sets and Solutions with Boundary Blow-up

    • Jean-François Le Gall
    Pages 89-109

  8. The Probabilistic Representation of Positive Solutions

    • Jean-François Le Gall
    Pages 111-128

  9. Lévy Processes and the Genealogy of General Continuous-state Branching Processes

    • Jean-François Le Gall
    Pages 129-149

  10. Back Matter

    Pages 151-163
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Keywords

About this book

In these lectures, we give an account of certain recent developments of the theory of spatial branching processes. These developments lead to several fas­ cinating probabilistic objects, which combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial dif­ ferential equations. Our first objective is to give a short self-contained presentation of the measure­ valued branching processes called superprocesses, which have been studied extensively in the last twelve years. We then want to specialize to the important class of superprocesses with quadratic branching mechanism and to explain how a concrete and powerful representation of these processes can be given in terms of the path-valued process called the Brownian snake. To understand this representation as well as to apply it, one needs to derive some remarkable properties of branching trees embedded in linear Brownian motion, which are of independent interest. A nice application of these developments is a simple construction of the random measure called ISE, which was proposed by Aldous as a tree-based model for random distribution of mass and seems to play an important role in asymptotics of certain models of statistical mechanics. We use the Brownian snake approach to investigate connections between super­ processes and partial differential equations. These connections are remarkable in the sense that almost every important probabilistic question corresponds to a significant analytic problem.

Reviews

"Concise and essentially self-contained… A very accessible text…written by a leading expert of the field… It provides a clear and precise presentation of several important aspects of the theory…developed over the recent years. There is no doubt that such a monograph will be used both by beginners to learn the theory and by experts as a reference text."

—Zentralblatt Math.

Authors and Affiliations

  • Département de Mathématiques, Ecole Normale Supérieure, Paris, France

    Jean-François Gall

Bibliographic Information

  • Book Title: Spatial Branching Processes, Random Snakes and Partial Differential Equations

  • Authors: Jean-François Gall

  • Series Title: Lectures in Mathematics. ETH Zürich

  • DOI: https://doi.org/10.1007/978-3-0348-8683-3

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 1999

  • Softcover ISBN: 978-3-7643-6126-6Published: 01 July 1999

  • eBook ISBN: 978-3-0348-8683-3Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: 163

  • Topics: Applications of Mathematics

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