几类广义分枝模型的研究

Markov branching process is one of the most important research fields in probability. It has widely applied in queueing system, population science, molecular biology and so on. Its essential characteristic is independence. However, in many practical cases, the independence events are effected by the interaction of two or more particles rather than by the particles individually. This dependency has attracted great interest, therefore, research on generalized branching model has become an important subject. In recent years, the project applicant has been studying this kind of complex system, on the basis of the classical markov process theory, fully considering the actual problem, cancelled the essence characteristic of the classical markov branching process - independence, which has been promoted to a class of branch can depict the actual phenomenon of species more complex branching model, and get a series of important achievements. This topic proposed by establishing multi-dimensional generalized branching model with immigration, put forward a new method, overcome the complexity of high-dimensional branching model, research the existence and uniqueness, the absorption probability, ergodicity, quasi-stationary distribution of the process and so on.

马氏分枝过程在排队系统、人口科学、分子生物学等领域应用广泛，是概率论研究领域的一个重要分支。经典分枝过程的本质特征是粒子的演变相互独立，然而，在许多实际模型中，粒子之间通常具有相互作用。这种依赖关系引起了人们的极大兴趣，因此，研究广义分枝模型便成为一个重要课题。近年来，本项目申请者一直致力于研究这种复杂系统，在经典马氏分枝过程理论的基础上，充分考虑实际问题，取消了经典马氏分枝过程的本质特征—独立性，将其推广到一类更能刻画实际分枝现象的多物种复杂分枝模型，并且取得了一系列重要成果。本课题拟通过建立带移民的高维广义分枝模型，提出新的方法，克服高维分枝模型的复杂性，进一步研究过程的存在唯一性、吸收概率、遍历性、衰减指数以及拟平稳分布等问题。