The Related Study Of Branching Processes Model In Random Environment

The classic branching processes was born in 1874. After the 95 years of development, Wilkinson and Smith [73] proposed a branching process in independent and identically distributed environment in 1969. Athreya K. B. [7] proposed a branching process in stationary ergodic environment combined with the ergodic theory in 1971. The branching process in random environment has great epoch-making significance for the development of the theory of branching processes. Thus it determines the general direction of the development of branching processes. The conclusion of the classical branching process in a random environment is being extended to the branching process in a random environment. When we do these promotional efforts, many obvious conclusions of classical branching processes is not necessarily founded in random environment. We need to combine much more ergodic theory, martingale theory, large deviations theory and random walk theory, etc. It is also the essence of these theories of branching processes in random environments.This paper obtained the following conclusions:1. Drawing on literature [1], we discuss the problem that extending Ratio Lemma in classical branching process to exchangeable environment branching process. Obtain Ratio Lemma in exchangeable environment branching process and necessary conditions of the summation series limit.2. Drawing on literature [55], we discuss the problem that extending the limit of E(szn| n+k< T) almost everywhere existence in classical subcritical branching processes to random environment branching processes. Obtain that Eζ (szn|n+k<T) convergent in distribution but not in probability.3. Drawing on special invariant measure of example in critical classical branching processes, we discuss issues about the existence of invariant measure in a random environment and get examples of subinvariant measure.4. Drawing on literature [8], we study the limit properties of branching process with immigration in random environment, get a result that Wn=Zn/∏j-0n-1 almost everywhere convergence to a non-degenerate random variable W. And then seek the specific forms of EζW andVar VarW. Describe the population explosion of supcritical branching process with immigration in random environment.5. Drawing on literature [39] and classification theorem of controlled branching processes, we study the extinction probability of controlled branching processes in random environment. Derive the necessary conditions for extinction and non-inevitable extinction of controlled branching process in random environment. And on this basis, obtain necessary and sufficient conditions for non-degenerate random variable limit.