Generalized Markov branching models
In this thesis, we first considered a modified Markov branching process incorporating both state-independent immigration and resurrection. After establishing the criteria for regularity and uniqueness, explicit expressions for the extinction probability and mean extinction time are presented. The criteria for recurrence and ergodicity are also established. In addition, an explicit expression for the equilibrium distribution is presented. We then moved on to investigate the basic properties of an extended Markov branching model, the weighted Markov branching process. The regularity and uniqueness criteria of such general structures are first established. There after closed expressions for the mean extinction time and conditional mean extinction time are presented. The explosion behaviour and the mean explosion time are also investigated. In particular, the Harris regularity criterion for ordinary Markov branching process is extended to a more general case of non-linear Markov branching process. Finally, we studied a new Markov branching model, the weighted collision branching process, and considered two special cases of this process. For this weighted collision branching process, some conditions of regularity and uniqueness are obtained and the extinction behaviour and explosion behaviour of the process are investigated. For the two special cases, a collision branching process and a general collision branching process with 2 parameters, the regularity and uniqueness criteria of the process are established and explicit expressions for extinction probability vector, mean extinction times, conditional mean extinction times and mean explosion time are all obtained.