Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes

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• 作者：J.-R. Chazottes ; P. Collet ; S. Méléard
• 关键词：Primary 92D25 ; Secondary 60J27 ; 60J28 ; 60J80 ; 47A75 ; 92D40
• 刊名：Probability Theory and Related Fields
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：164
• 期：1-2
• 页码：285-332
• 全文大小：753 KB
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• 作者单位：J.-R. Chazottes (1)
  P. Collet (1)
  S. Méléard (2)
  
  1. Centre de Physique Théorique, CNRS UMR 7644, Ecole polytechnique, 91128, Palaiseau Cedex, France
  2. Centre de Mathématiques Appliquées, CNRS UMR 7641, Ecole polytechnique, 91128, Palaiseau Cedex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Mathematical and Computational Physics
  Quantitative Finance
  Mathematical Biology
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Operation Research and Decision Theory
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-2064

文摘

We study a general class of birth-and-death processes with state space \({\mathbb {N}}\) that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a ‘carrying capacity’ \(K\). When \(K\) is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of \(K\), for large \(K\). We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when \(K\) is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process. Mathematics Subject Classification Primary 92D25 Secondary 60J27 60J28 60J80 47A75 92D40