Multiple scale analysis of spatial branching processes under the Palm distribution
type: Item
instance of: scholarly article
title: Multiple scale analysis of spatial branching processes under the Palm distribution
zbMATH Open document ID: 1010.60077
author: Anita Winter
published in: Electronic Journal of Probability
publication date: 2003-02-13T00:00:00Z
full work available at URL: https://eudml.org/doc/122586
full work available at URL: paper13.abs.html
review text: The paper deals with the continuous-time critical binary branching random walk on \(\mathbb{Z}^2\) where the random walk kernel is assumed to have a finite second moment, and with its continuous mass analogue, a linearly interacting critical Feller's branching diffusion on \(\mathbb{Z}^2\) (a particular super-random walk). Via log-Laplace transition functionals, both models have the same deterministic dual process, the solution of a ``diffusion''-reaction (or log-Laplace) equation on \(\mathbb{Z}^2\) with quadratic reaction term. Two well-known concepts are used to describe the formation of clusters in detail: (i) the Palm distribution approach, and (ii) multiple space-time scaling. It is shown that both approaches commute. Technically, this leads to tree indexed branching systems which are completely characterized by their time-space moments. These moments are described effectively and explicitly.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 1868592
zbMATH Keywords: infinite particle system
zbMATH Keywords: superprocess
zbMATH Keywords: interacting diffusion
zbMATH Keywords: clustering
zbMATH Keywords: Palm distribution
zbMATH Keywords: grove indexed systems of diffusions
zbMATH Keywords: grove indexed systems of branching models
zbMATH Keywords: Kallenberg's backward tree
MaRDI profile type: Publication
DOI: 10.1214/EJP.V7-112
description: scientific article; zbMATH DE number 1868592