Mutually catalytic branching in the plane: Infinite measure states
type: Item
instance of: scholarly article
title: Mutually catalytic branching in the plane: Infinite measure states
zbMATH Open document ID: 1016.60075
author: Donald A. Dawson
author: Alison Etheridge
author: Klaus Fleischmann
author: Edwin Perkins
author: Jie Xiong
author: Leonid Mytnik
published in: Electronic Journal of Probability
publication date: 2003-02-13T00:00:00Z
full work available at URL: https://eudml.org/doc/122772
review text: The authors construct a pair of infinite-measure-valued processes as a model for two-type population in \(\mathbb R^{2}\) which undergoes diffusion and branching. It is a rescaled limit of a corresponding \(\mathbb Z^{2}\)-lattice model studied by \textit{D. A. Dawson} and \textit{E. A. Perkins} [Ann. Probab. 26, 1088-1138 (1998; Zbl 0938.60042)]. The branching rate of each type of population is proportional to the local density of the other type, whereas the collision rate is sufficiently small compared with the diffusion rate. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of two populations. Local extinction of one type is shown (in law in the long-term limit) and the surviving population is uniform with random intensity. Uniqueness questions are treated by \textit{D. A. Dawson}, \textit{K. Fleischmann}, \textit{L. Mytnik}, \textit{E. A. Perkins} and \textit{J. Xiong} [Ann. Inst. Henri Poincaré, Probab. Stat. 39, 135-191 (2003; Zbl 1016.60091)].
reviewed by: Mihai Gradinaru
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 1868594
zbMATH Keywords: measure-valued branching
zbMATH Keywords: interactive and state-dependent branching
zbMATH Keywords: collision measure
zbMATH Keywords: martingale problem
zbMATH Keywords: segregation and coexistence of types
zbMATH Keywords: long-term behaviour
Wikidata QID: Q60500197
MaRDI profile type: Publication
DOI: 10.1214/EJP.V7-114
description: scientific article; zbMATH DE number 1868594