Spatial branching processes, random snakes and partial differential equations空间分支的过程、任意蛇和部份微分方程
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作者： Jean-Francois Le Gall 著
出 版 社：
出版时间： 1999-9-1字数：版次： 1页数： 162印刷时间： 1999/09/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783764361266包装： 平装内容简介
The text includes a presentation of the measure-valued branching processes also called superprocesses and of their basic properties. In the important quadratic branching case, the path-valued process known as the Brownian snake is used to give a concrete and powerful representation of superprocesses. This representation is applied to several connections with a class of semilinear partial differential equations. On the one hand, these connections give insight into properties of superprocesses. On the other hand, the probabilistic point of view sometimes leads to new analytic results, concerning for instance the trace classification of positive solutions in a smooth domain. An important tool is the analysis of random trees coded by linear Brownian motion. This includes the so-called continuum random tree and leads to the fractal random measure known as ISE, which has appeared recently in several limit theorems for models of statistical mechanics. This book is intended for postgraduate students and researchers in probability theory. It will also be of interest to mathematical physicists or specialists of PDE who want to learn about probabilistic methods. No prerequisites are assumed except for some familiarity with Brownian motion and the basic facts of the theory of stochastic processes. Although the text includes no new results, simplified versions of existing proofs are provided in several instances.
目录
Foreword
Frequently Used Notation
Chapter Ⅰ An Overview
Ⅰ.1 Galton-Watson processes and continuous-state branching processes
Ⅰ.2 Spatial branching processes and superprocesses
Ⅰ.3 Quadratic branching and the Brownian snake
Ⅰ.4 Some connections with partial differential equations
Ⅰ.5 More general branching mechanisms
Ⅰ.6 Connections with statistical mechanics and interacting particle systems
Chapter Ⅱ Continuous-state Branching Processes and Superprocesses
Ⅱ.1 Continuous-state branching processes
Ⅱ.2 Superprocesses
Ⅱ.3 Some properties of superprocesses
Ⅱ.4 Calculations of moments
Chapter Ⅲ The Genealogy of Brownian Excursions
Ⅲ.1 The It5 excursion measure
Ⅲ.2 Binary trees
Ⅲ.3 The tree associated with an excursion
Ⅲ.4 The law of the tree associated with an excursion
Ⅲ.5 The normalized excursion and Aldous' continuum random tree
Chapter Ⅳ The Brownian Snake and Quadratic Superprocesses
Ⅳ.1 The Brownian snake
Ⅳ.2 Finite-dimensional marginals of the Brownian snake
Ⅳ.3 The connection with superprocesses
Ⅳ.4 The case of continuous spatial motion
Ⅳ.5 Some sample path properties
Ⅳ.6 Integrated super-Brownian excursion
Chapter V Exit Measures and the Nonlinear Dirichlet Problem
Ⅴ.1 The construction of the exit measure
Ⅴ.2 The Laplace functional of the exit measure
Ⅴ.3 The probabilistic solution of the nonlinear Dirichlet problem
Ⅴ.4 Moments of the exit measure
Chapter Ⅵ Polar Sets and Solutions with Boundary Blow-up
Ⅵ.1 Solutions with boundary blow-up
Ⅵ.2 Polar sets
Ⅵ.3 Wiener's test for the Brownian snake
Ⅵ.4 Uniqueness of the solution with boundary blow-up
Chapter Ⅶ The Probabilistic Representation of Positive Solutions
Ⅶ.1 Singular solutions and boundary polar sets
Ⅶ.2 Some properties of the exit measure from the unit disk
Ⅶ.3 The representation theorem
Ⅶ.4 Further developments
Chapter Ⅷ L4vy Processes and the Genealogy of General Continuous-state Branching Processes
Ⅷ.1 The discrete setting
Ⅷ.2 LSvy processes
Ⅷ.3 The height process
Ⅷ.4 The exploration process
Ⅷ.5 Proof of Theorem 2
Bibliographical Notes
Bibliography
Index
