Superprocesses in random environments
type: Item
instance of: scholarly article
title: Superprocesses in random environments
zbMATH Open document ID: 0874.60041
author: Leonid Mytnik
published in: The Annals of Probability
publication date: 1997-11-02T00:00:00Z
review text: The paper establishes the weak convergence of a sequence of branching particle systems whose branching mechanism is affected by a random environment. The environment is specified by a sequence \(\xi_k\), \(k=1,2,\dots\), of independent, identically distributed zero-mean random fields on the space \(E\) in which the particles live. The \(n\)th particle system \(X^n_t\), \(t\geq 0\), starts at time \(t=0\) with \(K_n\) particles \((K_n\sim n)\) and during any time interval \((k/n,(k+1)/n)\), \(k=0,1,2,\dots\), all existing particles move independently of each other in accordance with a Feller process on \(E\). Conditionally on \(\xi^{(n)}_k\) (where \(k\geq 1\) and \(\xi^{(n)}_k\) is \(\xi_k\) truncated at \(\pm\sqrt n\)), each particle existing at time \(k/n\) either splits into two particles or dies, with probabilities \({1\over 2}+{1\over 2\sqrt n} \xi^{(n)}_k(x)\) and \({1\over 2}-{1\over 2\sqrt n} \xi^{(n)}_k(x)\), respectively, independently of other particles. Here \(x\) denotes the location of the particle. The author proves that if \(X^n_0\) converges, then the sequence \(X^n\), \(n=1,2,\dots\), converges weakly to a measure-valued superprocess which is the unique solution of a certain martingale problem.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 986792
zbMATH Keywords: weak convergence
zbMATH Keywords: branching particle systems
zbMATH Keywords: Feller process
zbMATH Keywords: measure-valued superprocess
zbMATH Keywords: martingale problem
reviewed by: F. Papangelou
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1041903212
OpenAlex ID: W2085564755
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DOI: 10.1214/AOP/1041903212
description: scientific article; zbMATH DE number 986792