Age-dependent branching processes in random environments
Age-dependent branching processes in random environments
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DOI：
10.1007/s11425-008-0065-4
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发表时间：
2008-09
期刊：
影响因子：
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通讯作者：
Yingqiu Li;Quansheng Liu
中科院分区：
文献类型：
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作者：
Yingqiu Li;Quansheng Liu
考虑随机环境中的年龄相关分支过程。环境由随机变量的平稳和遍历序列n =（n = 0，n = 1，...）表示。在给定的环境条件下，该过程是一个非齐次的Galton-Watson过程，其第n代粒子的寿命分布为G（？n），并且在？n上按概率律p（？n）独立地再生新粒子.令Z（t）为在时间点存活的粒子数。我们首先通过一个函数方程找到了Z（t）的条件概率母函数的一个特征，并通过与嵌入的Galton-Watson过程的比较，得到了该过程几乎必然灭绝的一个判据.通过研究随机环境中的更新方程，得到了条件均值E_∞ Z（t）和全局均值E_∞ Z（t）的表达式，并给出了它们的指数增长率.
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequenceξ= (ξ0,ξ1,…) of random variables. Given an environmentξ, the process is a non-homogenous Galton-Watson process, whose particles inn-th generation have a life length distributionG(ξn) on ℝ+, and reproduce independently new particles according to a probability lawp(ξn) on ℕ. LetZ(t) be the number of particles alive at timet. We first find a characterization of the conditional probability generating function ofZ(t) (given the environmentξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional meanEξZ(t) and the global meanEZ(t), and show their exponential growth rates by studying a renewal equation in random environments.