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Spatial Branching Processes, Random Snakes and Partial Differential Equations

  • Jean-François Le Gall

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

Table of contents

  1. Front Matter
    Pages i-ix
  2. Jean-François Le Gall
    Pages 1-20
  3. Jean-François Le Gall
    Pages 21-40
  4. Jean-François Le Gall
    Pages 41-51
  5. Jean-François Le Gall
    Pages 53-74
  6. Jean-François Le Gall
    Pages 75-88
  7. Jean-François Le Gall
    Pages 89-109
  8. Jean-François Le Gall
    Pages 111-128
  9. Back Matter
    Pages 151-163

About this book

Introduction

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.

Keywords

PDE Probability theory Stochastics classification mechanics partial differential equation

Authors and affiliations

  • Jean-François Le Gall
    • 1
  1. 1.Département de MathématiquesEcole Normale SupérieureParisFrance

Bibliographic information