Markov branching processes with immigration-migration and resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.

Markov branching processes with killing and resurrection

In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is honest and thus unique. We then further show that this honest Feller minimum process is not only positive recurrent but also strongly ergodic. The generating function of the important stationary distribution is explicitly expressed. For the interest of comparison and completeness, the results of the Markov branching processes with killing and instantaneous resurrection are also briefly stated. A new result regarding strong ergodicity of this difficult case is presented. The birth and death process with killing and resurrection together with another example is also analyzed.

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

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

Generalized Markov interacting branching processes

We consider a very general interacting branching process which includes most of the important interacting branching models considered so far. After obtaining some key preliminary results, we first obtain some elegant conditions regarding regularity and uniqueness, Then the extinction vector is obtained which is very easy to be calculated. The mean extinction time and the conditional mean extinction time are revealed.The mean explosion time and the total mean life time of th, processes are also investigated and resolved.

A new approach in analyzing extinction probability of Markov branching process with immigration and migration

We use a new approach to consider the extinction properties of the Markov branching process with immigration and migration recently discussed by Li and Liu [Sci. China Math., 2011, 54： 1043-1062]. Some much better explicit expressions are obtained for the extinction probabilities of the subtle super-interacting case.

Decay parameter and related properties of 2-type branching processes

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.

Decay parameter and related properties of n-type branching processes

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions for n-type Markov branching processes on the basis of the 1-type Markov branching processes and 2-type Markov branching processes. Investigating such behavior is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for n-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 = Zn+ \ 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. λC -invariant measures and quasi-stationary distributions for the process on C are presented.

General collision branching processes with two parameters

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