Research On Markov Branching Models With Instantaneous Resurrection

In this thesis,we investigate two classes of Markov branching models with immigration and instantaneous resurrection,which involves quadratic weighted Markov model and 2-type Markov branching model.We further research the existence criterion,uniqueness criterion and recurrent properties of these models.The research content is mainly divided into the following two parts:The first part analyzes a quadratic weighted Markov branching model with immigration and instantaneous resurrection.Firstly,the definition of this model is given and its relationship with other similar models is analyzed.We first proved four lemmas as the theoretical basis.Then,using the resolvent decomposition theorem,it is proved that the process does not exist when the resurrected rate is finite.Furthermore,based on the transformation relationship between the restrained matrix and the absorption matrix,the necessary and sufficient condition for the existence of the process is obtained,and some equivalent conditions for easy verification are stated.Regarding the uniqueness of processes,it has been proven that for a given function,there are infinitely many processes corresponding to it,but only one of them is honest.The construction method for this honest process is provided.We discussed the recurrence of this honest process,and provided the equation that the stationary distribution satisfies.The second part studies a 2-type Markov branching model with immigration and instantaneous resurrection.Firstly,the evolution law of the system is presented,and the definition of the model is provided.Then assume some proper conditions to avoid the process degenerate.We first proved two lemmas as the theoretical basis,and then used the resolvent decomposition theorem to prove that the process does not exist when the resurrected rate is finite.By using the conversion formula between the restrained matrix and the absorption matrix,the dimensional of the independent variables in the Kolmogorov forward equation is reduced,and a system for judging the existence of the process was established.Regarding the uniqueness of the process,it is proved that there are infinitely many 2TBP-IIR corresponding to a given function,and exactly one of them is non interruptive.The construction method of this honest process is also provided.Finally,the recurrence of honest processes is analyzed,and the equations that the stationary distribution satisfies and the equations that the stationary distribution satisfies when there is no immigration are given.