高维Markov碰撞分枝模型的衰减性研究

Markov branching process is a kind of special stochastic process, which plays an important role in probability and its application. The essential characteristic of classical Markov branching processes is independence. However, in many practical cases, the independence events are affected by the interaction of two or more particles. Therefore, the research on generalized branching model has become an important subject. Many achievements and satisfactory progress have been achieved in the study of generalized branching model. But for the high dimensional interacting branching model, because of the complexity of many species in the system, it is difficult to describe the decay properties of these models. In this project, we mainly study the high-dimensional branching model with interaction between particles and the high-dimensional collision-branch-migration model with multiple particles in the system, cancel the essential characteristics of the classical Markov branching process-independence, then the classical Markov branching process is extended to a class of high dimensional model can better describe the actual collision branching phenomenon. By studying the deep properties of generating function family and using new methods and techniques, we overcome the complexity of high-dimensional branching models, solve the extinction probability and the mean extinction time of processes, further establish the recurrence and ergodicity criteria of models, and give the ergodic parameter (or decay parameter) and the stationary distribution (or quasi-stationary distribution), at the same time, study relate properties of the branching-queuing system, in order to understand these models theoretically.

Markov分枝过程是研究得相当深入，而且还在蓬勃发展的一类重要随机过程。其本质特征是粒子的独立性，然而在许多实际问题中，粒子之间通常是紧密相联的，因而广义分枝模型的研究越来越引起人们的关注，并已取得丰富的成果和令人满意的进展。但这些工作主要集中在系统中只有一种粒子或粒子间相互独立的情形，而对于高维交互分枝模型，由于系统中多个物种的复杂性，给这些模型衰减性质的刻画带来了本质困难。本项目主要研究粒子间具有交互作用的高维分枝模型及系统中具有多种粒子的高维碰撞-分枝-移民模型，取消了经典马氏分枝过程的本质特征-独立性，通过研究发生函数族的深层性质，利用新的方法和技巧，克服高维分枝模型的复杂性，解决过程的灭绝概率及平均灭绝时间等问题，进一步建立模型的常返性和遍历性准则，给出遍历指数（或衰减指数）及平稳分布（或拟平稳分布），同时研究分枝-排队系统的相关性质，旨在从理论上理解这些模型所描述的自然现象。

Markov分枝过程是研究得相当深入，而且还在蓬勃发展的一类重要随机过程。其本质特征是粒子的独立性，然而在许多实际问题中，粒子之间通常是紧密相联的，因而广义分枝模型的研究越来越引起人们的关注，广义分枝模型的研究便成为一个重要课题。.本课题建立了连续时间的马尔可夫分枝模型，我们在已有工作的基础上问题，研究这一模型的下偏差问题，即种群或粒子的增长低于平均速度时，这一小概率事件的极限行为，试图从概率意义上解释这一模型的渐近性质，为后续大偏差的研究及应用提供了理论依据。对于上临界的连续时间的马尔可夫分枝-移民系统，引入了模型的发生函数，建立了全新的泛函方程，并且利用Kolmogorov向前方程构造了发生函数对应的常微分方程，讨论了其总人口数调和矩的极限行为。其次，作为应用，我们得到了Lotka-Nagaev估计量的大偏差，进一步完善了连续时间大偏差的理论体系。在此基础上，进一步讨论了模型的自正则化大偏差问题，将离散时间的马尔可夫分枝过程的偏差问题研究成功推广到了连续时间模型上，这是一个值得期待的课题，具有新的应用前景。