Existence And Uniqueness Of Quasi-stationary Distributions In Collision Branching Processes

At present,quasi-stationary distributions of Markov processes plays an impor-tant role in biology,physics and engineering science.In this paper,the existence and uniqueness of quasi-stationary distributions in collision branching processes are studied.It is mainly divided into the following three parts:The first part introduces the research background,history and the main work of this paper.The second part is the preliminary knowledge,which explains the basic knowledge of Markov chain and the definition of decay parameter of process with absorption state,and also introduces the concept of quasi-stationary distribution of process.The third part is the core part of this paper.Firstly,the model of collision branching processes and its related properties are introduced.Under the premise of probability 1 extinction of the model,it is proved that the mean extinction time is uniformly bounded,that is,₄∈)₄)<∞.It is also proved that the decay parameter of collision branch-ing processes is greater than 0,and the collision branching processes has a unique quasi-stationary distribution,and the process converges to a unique quasi-stationary distribution from any initial distribution.