Ebook: Spatial Branching Processes, Random Snakes and Partial Differential Equations

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.

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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.

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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.
Content:
Front Matter....Pages i-ix
An Overview....Pages 1-20
Continuous-state Branching Processes and Superprocesses....Pages 21-40
The Genealogy of Brownian Excursions....Pages 41-51
The Brownian Snake and Quadratic Superprocesses....Pages 53-74
Exit Measures and the Nonlinear Dirichlet Problem....Pages 75-88
Polar Sets and Solutions with Boundary Blow-up....Pages 89-109
The Probabilistic Representation of Positive Solutions....Pages 111-128
L?vy Processes and the Genealogy of General Continuous-state Branching Processes....Pages 129-149
Back Matter....Pages 151-163

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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.
Content:
Front Matter....Pages i-ix
An Overview....Pages 1-20
Continuous-state Branching Processes and Superprocesses....Pages 21-40
The Genealogy of Brownian Excursions....Pages 41-51
The Brownian Snake and Quadratic Superprocesses....Pages 53-74
Exit Measures and the Nonlinear Dirichlet Problem....Pages 75-88
Polar Sets and Solutions with Boundary Blow-up....Pages 89-109
The Probabilistic Representation of Positive Solutions....Pages 111-128
L?vy Processes and the Genealogy of General Continuous-state Branching Processes....Pages 129-149
Back Matter....Pages 151-163
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