On the quasi-stationary distribution for some randomly perturbed transformations of an interval

• 成果类型: Article
• 署名作者: Klebaner, FC; Lazar, J; Zeitouni, O
• 署名单位: University of Melbourne; University of Melbourne; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 1998
• 页码: 300-315
• 关键词: branching-processes; threshold

摘要

We consider a Markov chain X-n(epsilon) obtained by adding small noise to a discrete time dynamical system and study the chain's quasi-stationary distribution (qsd). The dynamics are given by iterating a function f: I --> I for some interval I when f has finitely many fixed points, some stable and some unstable. We show that under some conditions the quasi-stationary distribution of the chain concentrates around the stable fixed points when epsilon --> 0. As a corollary, we obtain the result for the case when f has a single attracting cycle and perhaps repelling cycles and fixed points. In this case, the quasi-stationary distribution concentrates on the attracting cycle. The result applies to the model of population dependent branching processes with periodic conditional mean function.