随机环境中马氏链与多型分枝过程

This project mainly studies the properties of multitype branching processes in random environments. In the supercritical case, we consider the asymptotical properties of the processes, including the existence of moments and weighted moments, large deviation principals and the convergence rate of the related martingale. In the critical or subcritical case, we study the asymptotical properties of the survival probability of the whole process or of some special type particles, and the distribution of the population size conditioned on the survival, etc. . Next, in connection with our finished projects, we further study in depth the properties of other concrete models of Markov chains in random environments, including: bisexual branching processes in random environments, (single type) branching processes, branching random walks, age-dependent branching processes, randomly indexed branching processes, Sevast'yanov branching models, single birth chains, controlled branching processes with random control function, renewal processes and superprocesses. Meanwhile, we will also continue to consider some important problems about the general theory of Markov chains in random environments. Moreover, we shall also try to apply the new methods and results on branching processes in random environments to study some important related problems. . Some research topics involved in this project are still blank both at home and abroad. In the study of multitype branching processes in random environments, we have got some interesting new methods and theoretical results, which can be applied to the study of related research fields. This lays for us a good foundation for the study of multi-type branching processes in random environments and related topics.

主要研究随机环境中多型分枝过程的性质,对上临界情形,研究过程的渐近性质,包括大偏差原理、矩和加权矩的存在性、自然鞅的收敛速率等.对临界或者下临界情形,研究过程或特定型粒子之生存概率的渐近性质,以及在存活条件下种群数量的分布等.. 进一步深入研究随机环境中马氏链其它具体模型,如随机环境中两性分枝过程、(单型)分枝过程、分枝随机游动、依赖寿命的分枝过程、随机指标分枝过程、单生链、受控分枝过程、更新过程、超过程等的性质.继续考虑随机环境中马氏链一般理论的一些重要问题,希望在其中一些问题上有重要突破.尝试把随机环境中分枝过程研究中的新方法和有关理论成果应用于有意义的相关问题的研究中.. 本项目涉及的一些研究方向尚属国内外的空白,而我们对随机环境中分枝过程研究已有一定的基础,有望在随机环境中多型分枝过程及相关课题的研究上取得好的成果.

本项目最重要的进展是随机环境中多型分枝过程基本鞅的发现,这将开辟一系列重要问题的研究途径:将极限随机变量非退化的定理,以及关于种群规模的规范化过程,从确定环境推广到随机环境情形; 将随机环境中单型分枝过程的大偏差等结果推广到多型的情形.. 本项目研究了随机环境中分枝过程的下临界与临界的概念、比率定理、调和矩、大偏差、Berry-Esseen界和Cramer大偏差展式,分枝随机游动中心极限定理的一阶和二阶展式,时间随机环境中分枝随机游动的渐近性质、粒子分布的二阶和三阶渐近展开、中心极限定理的精确收敛速率,随机环境中Mandelbrot瀑布调和矩、大偏差和中偏差原理,随机环境中受控分枝过程的极限定理,随机环境中具有移民的上临界分枝过程的极限定理,分枝型重载轮询网络的渐近性质等.. 在此基础上,根据国内外随机环境中马氏链与多型分枝过程研究发展情况及项目进展情况,我们还组织研究力量讨论了在风险理论和生物统计等其它研究领域中的应用.本项目的研究进一步完善了马氏过程的整个理论体系.