Spatial Branching Processes, Random Snakes and Partial Differential Equations

Jean-Francois Le Gall

ISBN 10: 3764361263 / ISBN 13: 9783764361266

Edité par Birkhäuser Basel, Birkhäuser Basel Jul 1999, 1999

A propos de cet article

Neuware -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.Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin 176 pp. Englisch.

N° de réf. du vendeur 9783764361266

Détails bibliographiques

• Titre: Spatial Branching Processes, Random Snakes and Partial Differential Equations
• Éditeur: Birkhäuser Basel, Birkhäuser Basel Jul 1999
• Date de parution: 1999
• Langue: anglais
• ISBN à 10 chiffres: 3764361263
• ISBN à 13 chiffres: 9783764361266
• Reliure: Taschenbuch
• État de l'article: Neu

A propos de ce titre

Synopsis

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.