The Research For Property Of Related About The Controlled Branching Process

In 1873 Galton and Waston published parpers respectively about the new random process model established in the issue on surname inheritance and lineage extinction of the British aristocracy.It is just these two papers laid the theoretical foundation for the classic branching processes. Research on branching processes is one active and fruitful branch in the researching field of probability theory. Galton-Waston process (GW-process) with discrete states in discretetime is the simplest model of branching processes which describe the evolution of particle groups and has basic assumption each particle produces descendant independently with the same probability distribution. But in reality, the reproduction between the individuals is affected by many factors, so we introduced various types of branching process model, from homogeneous branching processes to non-homoge- neous branching process, from single type branching processes to multi- type branching processes, from classical branching processes to branching processes in random environments.Among the many expansion models, expansion from the controlled branching processes in determinate environment to the controlled branching processes in random environment is a widespread concerned topic over the past decade that a number of scholars had an in-depth research on.The research on the expansion from branching processes in determinate environment to branching processes in random environment has a large complete theoretical system which conducts an in-depth study of innovation in four major areas. We will generalize the controlled branching processes model from the determinate environment to random environment basing on the above-mentioned four aspects. Firstly, the model of the controlled branching processes in determinate environment is extended to the model of controlled branching processes in random environment. Secondly, the relationships between the probability generating function and the related moment about controlled branching processes in random environment are considered acoording to the relationships in the determinate environment. Thirdly extinction probability of the controlled branching processes in random environment is discussed. Finally the asymptotic properties is studied.