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General collision branching processes with two parameters

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Science China Mathematics 2009年第7期52卷 1546-1568页

作者： CHEN AnYue LI junPing Department of Mathematical Sciences the University of LiverpoolLiverpoolL69 7ZLUK Department of Statistics and Actuarial Science University of Hong KongPokfulam RoadHong Kong School of Mathematical Science and Computing Technology Central South UniversityChangsha 410075China

A new class of branching models,the general collision branching processes with two parameters,is considered in this *** such models,it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing *** and uniqueness criteria are firstly *** expressions are then obtained for the extinction probability vector,the mean extinction times and the conditional mean extinction *** explosion behavior of these models is investigated and an explicit expression for mean explosion time is *** mean global holding time is also *** is revealed that these properties are substantially different between the super-explosive and sub-explosive cases.

关键词： Markov branching process general collision branching process uniqueness extinction probabilities mean extinction time mean explosion time 60j27 60j80

Age-dependent branching processes in random environments

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Science China Mathematics 2008年第10期51卷 1807-1830页

作者： LI YingQiu LIU QuanSheng College of Mathematics and Computing Science Changsha University of Science and TechnologyChangsha 410076China LMAM Universit′e de Bretagne SudCampus de TohannicBP57356017 VannesFrance

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) 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 mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.

关键词： age-dependent branching processes random environments probability generating function integral equation extinction probability exponential growth rates of expectation and conditional expectation random walks and renewal equation in random environments renewal theorem 60j80 60K37 60K05

Some scaled limit theorems for an immigration super-Brownian motion

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Science China Mathematics 2008年第2期51卷 203-214页

作者： ZHANG Mei School of Mathematical Sciences,Beijing Normal University,Beijing 100875,China 1. School of Mathematical Sciences Beijing Normal University Beijing 100875 China

In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue *** first prove a functional central limit theorem,and then study the l... 详细信息

关键词： super-Brownian motion with immigration functional central limit theorem large deviation moderate deviation 60j80 60F05

Existence,uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

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Science China Mathematics 2008年第7期51卷 1266-1286页

作者： LI junPing1 & CHEN AnYue2,3 1 School of Mathematical Science and Computing Technology,Central South University,Changsha410075,China 2 Department of Mathematical Sciences,The University of Liverpool,Liverpool,L69 7ZL,UK 3 Department of Statistics and Actuarial Science,University of Hong Kong,Pokfulam Road,Hong Kong,China 1. School of Mathematical Science and Computing Technology Central South University Changsha 410075 China 2. Department of Mathematical Sciences The University of Liverpool Liverpool L69 7ZL UK 3. Department of Statistics and Actuarial Science University of Hong Kong Pokfulam Road Hong Kong China

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous *** existence criterion of the process is firstly *** prove that if the sum of the resurrection rates is finite,then there does not exist any *** existence criterion is then established when the sum of the resurrection rates is *** equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter *** uniqueness criterion for such process is also *** prove that although there exist infinitely many of them,there always exists a unique honest process for a given *** unique honest process is then *** ergodicity property of this honest process is analysed in *** prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.

关键词： Markov branching process immigration resurrection recurrence ergodicity 60j27 60j80

Decay parameter and related properties of 2-type branching processes

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Science China Mathematics 2009年第5期52卷 875-894页

作者： LI junPing School of Mathematical Science and Computing Technology, Central South University, Changsha 410075, China 1. School of Mathematical Science and Computing Technology Central South University Changsha 410075 China

We consider the decay parameter, invariant measures/vectors and quasi-stationary dis- tributions for 2-type Markov branching processes. Investigating such properties is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for 2-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Z+2 \ 0. It is shown that this λC can be directly obtained from the generating functions of the corresponding q-matrix. Moreover, the λC-invariant measures/vectors and quasi-distributions of such processes are deeply considered. A λC-invariant vector for the q-matrix (or for the process) on C is given and the generating functions of λC-invariant measures and quasi-stationary distributions for the process on C are presented.

关键词： 2-type Markov branching process decay parameter invariant measures invariant vectors quasi-stationary distributions 60j27 60j80

On the weak convergence of super-Brownian motion with immigration

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Science China Mathematics 2009年第9期52卷 1875-1886页

作者： ZHANG Mei School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China School of Mathematical Sciences Beijing Normal University Beijing 100875 China

We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting proc... 详细信息

关键词： super-Brownian motion with immigration occupation time process central limit theorem tightness 60j80 60F05