With Immigration, Remove And Transient Rescue Of Markov Branching Process Existence And Ergodicity,

We consider a modified Markov branching process incorporating with both state-independent immigration-migration 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 was 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 analyzed in detail. We proved that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.In chapter 1, we mainly give out the introduction of this paper. The historical background of the research problem and the significance of this thesis even the research status are expounded. Markov process is a kind of important stochastic process, and the branching process is a special kind of Markov process, so it has many scholars and experts. In this chapter, the evolution of branching process that from classical branching process to the complex problems research considering the immigration under branch structure are reviewed. And the main job of this paper is briefly introduced.In chapter 2, some knowledge needed in this thesis is summarized. Much knowledge relating the Markov process needs to be given before researching. Therefore, in the second chapter, we have defined the Markov process and given out the q-matrix of process, and both the transition function and the Laplace transform of transition function are also introduced. The most important things are that some generating functions and their properties are presented. At last, we give out the absorbing probability of the MB-IMIR which plays an important role in subsequent chapters'research.Using Resolvent Decomposition Theorem, we give out the proof of the existence and uniqueness of the Markov Branching Process with Immigration-Migration and Instantaneous Resurrection in chapter 3. The construction problem of the honest MB-IMIR is also resolved.In chapter 4, we mainly discuss the recurrence and ergodicity of the researching process. Firstly, the recurrence and ergodicity criteria for MB-IMIR are established. Secondly, the expression of the equilibrium distribution about the process is given.