Some Problems Of Galton-watson Branching Processes With Random Environments

Branching process theory is a classical method to analog evolution and degradation of population, which is formed gradually in the research on the extinction of English aristocratic surnames and families by Galton and Watson in 1873. The branching process in random environments was originally posed by Wilkinson in 1971 and Smith in 1967, which enriched modern branching process theory greatly. Compared with the·classical branching process, more accurate and more profound results could be obtained when the branching process in random environments is applied to deal with some molds similar to simplified population model, such as multiplication, particle fission, nuclear chain reaction, gene mutation, epidemic transmission, the undulate phenomenon of queuing theory and so on. At the same time, with the availability of hi-speed information technology, the theory of branching processes in random environments would have wider and newer applications in combinatorics, biology, demography, ecology and epidemiology et al. Some fundamental problems have been studied in this thesis, which consists of five chapters.Chapter one:Introduction. Research backgrounds and developments of branching processes in random environments have been introduced. Rigorous mathematical description of the model has been given and main results of this thesis.Chapter two:Limit Theorems. The convergence of normalized branching processes in random environments and index limit law have been studied.Chapter three:Stationary Measures. Ratio theorem of branching processes with independent and identically distributed environments has been studied. Existence and uniqueness of stationary measures for this models has been proved.Chapter four:Extinction time and extinction probability of Branching Processes in Random Environments. Some properties of extinction time and extinction probability of branching processes in random environments have been discussed.Chapter five:Randomly Indexed Branching Processes in Random Environments. The asymptotic formulas for the moments and none-extinction probabilities of randomly indexed branching processes in random environments have been given.Chapter six:The extinction probability and asymptotic formulas of contrulled branching processes with random environments. Markov property,the extinction probability and the asymptotic property of the contrulled branching processes with random environments have been discussed.