带拯救的广义非线性Markov分枝模型的稳定性

数学物理学报 ›› 2015, Vol. 35 ›› Issue (6): 1190-1206.

带拯救的广义非线性Markov分枝模型的稳定性

张利娜^1,2, 李俊平², 耿世锋¹

1 湘潭大学数学与计算科学学院湖南湘潭 411105;
2 中南大学数学与统计学院长沙 410075

收稿日期:2014-12-08 修回日期:2015-10-20 出版日期:2015-12-25 发布日期:2015-12-25
作者简介:张利娜,zln514@163.com;李俊平,jpli@mail.csu.edu.cn;耿世锋,sfgeng@xtu.edu.cn
基金资助:
国家自然科学基金(11371374,11301443)、教育部博士点基金(20110162110060,20124301120002)和湖南省自然科学基金项目(2015JJ3125)资助

The Stability Property of Generalized Nonlinear Markov Branching Models with Resurrection

Zhang Lina^1,2, Li Junping², Geng Shifeng¹

1 School of Mathematics and Computational Science, Xiangtan University, Hunan Xiangtan 411105;
2 School of Mathematics and Statistics, Central South University, Changsha 410075

Received:2014-12-08 Revised:2015-10-20 Online:2015-12-25 Published:2015-12-25

摘要/Abstract

摘要：

该文考虑了一类带拯救的广义非线性Markov分枝模型.首先讨论了带拯救的广义非线性Markov分枝q-矩阵发生函数的性质,通过发生函数给出了过程的正则性和唯一性判别准则,得到了没有拯救情形下过程的灭绝概率和平均灭绝时间,并讨论了带有拯救情形下模型的稳定性和遍历性,得到了过程的常返性和遍历性的充分必要条件.最后,给出了遍历情形下该过程的平稳分布.

关键词: 广义非线性Markov分枝模型, 拯救, 正则, 遍历

Abstract:

In this paper, the generalized non-linear Markov branching model with resurrection is considered. Some properties of the generating functions for generalized non-linear Markov branching q-matrix with resurrection are firstly investigated. By using the generating functions of the corresponding q-matrix, the criteria for regularity and uniqueness for such structure are firstly established, and the explicit expressions for the extinction probabilities and mean extinction times are presented. The stability properties and ergodicity of the model with resurrection are then investigated. The conditions for recurrence, ergodicity are obtained. An explicit expression for the equilibrium distribution is further presented.

Key words: Generalized non-linear Markov branching process, Resurrection, Regularity, Ergodicity

中图分类号:

O211.6

张利娜, 李俊平, 耿世锋. 带拯救的广义非线性Markov分枝模型的稳定性[J]. 数学物理学报, 2015, 35(6): 1190-1206.

Zhang Lina, Li Junping, Geng Shifeng. The Stability Property of Generalized Nonlinear Markov Branching Models with Resurrection[J]. Acta mathematica scientia,Series A, 2015, 35(6): 1190-1206.

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