Quasi-stationary distributions for structured birth and death processes with mutations

• 成果类型: Article
• 署名作者: Collet, Pierre; Martinez, Servet; Meleard, Sylvie; San Martin, Jaime
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Universidad de Chile; Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0297-4
• 发表日期: 2011
• 页码: 191-231
• 关键词: one-dimensional diffusions; CONVERGENCE; population; models; time

摘要

We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of the quasi-stationary distributions of the process conditioned on non-extinction. We first show the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.