| Markov branching process (MBPS) is a very important stochastic process in the process of branch, it has extensive application in many natural science and edge science, such as, medicine, queuing theory, modern physics, biology maths and so on. The paper mainly considers a special kind of branching process, that is the collision branching process with instantaneous resurrection. In this paper, we give the existence and uniqueness criterion of this process. At the same time, for a given q-matrix, there always exists an unique honest Q-process. This unique honest Q-process is then constructed. The ergodicity property of this honest Q-process is analyzed in detail. We prove that this honest Q-process is always ergodic. This paper mainly includes four parts:The first chapter is introduction. The introduction of this paper introduces the historical background of the research problem, the research status and the main content of this paper.In this chapter, we also introduce the generation and development process of the collision branching process, which lays a solid foundation for our future study.In the second chapter of this paper, some knowledge needed in this thesis is summarized. Mainly including:the basic concepts and properties of Markov chain and transition function; the definition of Q-matrix and its properties; the existence and uniqueness of Q-function; the basic concepts and properties of ergodic; the main content of Resolvent and Resolvent decomposition theorem. Finally we give out the definition of the collision branching process with instantaneous resurrection. At the same time, some generating functions and their properties are presented.The third chapter of this paper mainly considers the existence and uniqueness of this process. By using resolvent decomposition theorem, we give the proof of the existence and uniqueness of the collision branching process with instantaneous resurrection. The construction problem of this process is also resolved.The fourth chapter of this paper mainly discuss the recurrence and ergodicity of the collision branching process with instantaneous resurrection. In this chapter, we give the recurrence and ergodicity criteria for this honest process. |
