QUASI-STATIONARY DISTRIBUTIONS AND DIFFUSION MODELS IN POPULATION DYNAMICS

• 成果类型: Article
• 署名作者: Cattiaux, Patrick; Collet, Pierre; Lambert, Amaury; Martinez, Servet; Meleard, Sylvie; San Martin, Jaime
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Physics (INP); Sorbonne Universite; Universite Paris Cite; Universidad de Chile; Universidad de Chile
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP451
• 发表日期: 2009
• 页码: 1926-1969
• 关键词: one-dimensional diffusions; branching-process; markov-chains; CONVERGENCE

摘要

In this paper we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to -infinity at the origin, and the diffusion to have an entrance boundary at +infinity. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth-death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability 1. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near 0 and near +infinity for the existence of quasi-stationary distributions, as well as rate of convergence in the Yaglom limit and existence of the Q-process. We also show that, under these conditions, there is exactly one quasi-stationary distribution, and that this distribution attracts all initial distributions under the conditional evolution, if and only if +infinity is an entrance boundary. In particular, this gives a sufficient condition for the uniqueness of quasi-stationary distributions. In the proofs spectral theory plays an important role on L-2 of the reference measure for the killed process.