Spatial branching in random environments and with interaction
type: Item
instance of: scholarly article
zbMATH Open document ID: 1332.60003
DOI: 10.1142/8991
author: János Engländer
publication date: 2014-11-13T00:00:00Z
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60J85
Mathematics Subject Classification ID: 60K37
Mathematics Subject Classification ID: 92-02
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 6369222
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1142/8991
OpenAlex ID: W652691071
title: Spatial branching in random environments and with interaction
published in: Advanced Series on Statistical Science \& Applied Probability
review text: The book is concerned with several stochastic models involving spatial motion, branching, random environments and interaction between particles, in domains of \(\mathbb{R}^d\). The following is a description of the book chapters.NEWLINENEWLINEIn Chapter 1, the author calls several notions related to the subject of the book. This greatly facilitates reading of the subsequent chapters. In particular, some attention is paid to stochastic differential equations; martingale change of measure; Brownian motion and more general diffusions; branching processes, branching diffusions and superprocesses. In Chapter 2, the author investigates a dyadic branching diffusion \(Z\) which corresponds to the operator \(Lu+\beta(u^2-u)\) on a non-empty domain \(D\) in \(\mathbb{R}^d\), where \(L\) is a `smooth' second order elliptic differential operator on \(D\) and \(\beta\) is a non-zero Hölder continuous function. Let \(\lambda_c\) denote the generalized principal eigenvalue for the operator \(L+\beta\) on \(D\), i.e., \(\lambda_c\) is the infimum over all real \(\lambda\) for which there exists a positive function \(u\) such that \((L+\beta-\lambda)u=0\) on \(D\). The main result of the chapter, that the author calls the strong law of large numbers for the local mass of branching diffusions, asserts that when \(\lambda_c\in (0,\infty)\) and some additional assumptions hold, the random measures \(e^{-\lambda_c t}Z_t\) converge almost surely in the vague topology as \(t\to\infty\). The principal technical tool here is the `spine change of measure'. This is a generalization of the well-known and commonly used technique in the field of branching processes. Further, the a.s.\ and \(L_p\) convergence of certain martingales play an important role. The martingales can be seen as counterparts of Biggins' martingales which are closely related to branching random walks. In Chapter 3, the author gives examples of branching diffusions which satisfy the strong law of large numbers discussed in t…
reviewed by: Alexander Iksanov
description: scientific article; zbMATH DE number 6369222