On birth and death processes in symmetric random environment
type: Item
instance of: scholarly article
title: On birth and death processes in symmetric random environment
zbMATH Open document ID: 0587.60088
DOI: 10.1007/BF01010495
author: Kiyoshi Kawazu
author: Harry Kesten
published in: Journal of Statistical Physics
publication date: 1984-01-01T00:00:00Z
review text: We prove a limit theorem for a process in a random one-dimensional medium, which has been considered before as a model for hopping conduction in a disordered medium. To the edge between the two integers j and \((j+1)\) a rate \(\lambda_ j>0\) is attached. These \(\{\lambda_ j:\) j integral\(\}\) are taken as independent, identically distributed random variables, and represent the medium. For given values \(\lambda_ j\), X(t) is a Markov chain in continuous time which jumps from j to \((j+1)\) and from \((j+1)\) to j at the same rate \(\lambda_ j.\) We show that in many cases there exists normalizing constants \(\gamma\) (t) (which tend to \(\infty\) with t) such that the distribution of X(t)/\(\gamma\) (t), or more generally of the whole process \(\{X(st)/\gamma (t)\}_{s\geq 0}\), converges to a limit as \(t\to \infty\). The limit process is continuous and self-similar.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 3942673
zbMATH Keywords: random environment
zbMATH Keywords: birth and death process
zbMATH Keywords: invariance principle
zbMATH Keywords: self- similar
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01010495
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description: scientific article; zbMATH DE number 3942673