Persistent survival of one-dimensional contact processes in random environments
type: Item
instance of: scholarly article
title: Persistent survival of one-dimensional contact processes in random environments
zbMATH Open document ID: 0863.60098
published in: The Annals of Probability
publication date: 1997-06-10T00:00:00Z
review text: The authors consider a one-dimensional contact process in a random environment. It is defined as a Markov process with state space \(\{0,1\}^{\mathbb{Z}}\) with transitions \(1\to 0\) at site \(x\) with rate \(\delta(x)\) and \(0\to 1\) at site \(x\) with rate \(\varepsilon[\eta_t(x-1)+\eta_t(x+1)]\), where \(\eta_t\) is the state of the process. The focus of the article is on the event that the process \(\eta_t\) starting from the point \(\eta_0^{(j)}(x)=\begin{cases} 1, &x=j,\\ 0, &x\neq j,\end{cases}\) is not identically zero for every \(t>0\) (i.e. survives). In particular, it is proved that if \(u \mathbb{P}(-\log\delta(x)>u)\) tends to \(+\infty\) as \(u\to +\infty\), then \(\eta_t\) survives for every \(\varepsilon >0\). The discrete and continuous time processes are considered.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B44
zbMATH DE Number: 926219
zbMATH Keywords: survival
zbMATH Keywords: contact process in a random environment
zbMATH Keywords: discrete and continuous time processes
author: Charles M. Newman
author: S. B. Volchan
reviewed by: A. Khorunzhy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1042644723
OpenAlex ID: W2004950466
cites work: Discontinuity of the percolation density in one dimensional \(1/| x- y| ^ 2\) percolation models
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cites work: Q3976837
cites work: The survival of one-dimensional contact processes in random environments
DOI: 10.1214/AOP/1042644723
description: scientific article; zbMATH DE number 926219