广义分枝过程和粒子系统

Branching process is an important part of stochastic processes, which involves in probability theory, Markov processes, math physics, finance and insurance. General branching processes are the extensions of classical GW processes. Their structures lead to better mimic real life of the biological family tree and therefore of great theoretical and application significance. In recent decades, random trees integrate with branching processes intensively, and become a hot topic in probability research field. Particle system is the intersection of mathmatics and other related subjects. Many models, which generate from application subjects, have similar structures to various branching particle sysytems. Limit theory is clearly one of the main topics in probability study. Approximation in Markov chain is also the front research subject. There are still many open problems in general branching processes and related particle systems. We are devoted to: the construction of general branching processes and limit theorems with the help of random tree; limits of SPR- tree-valued processes; conditional limit theorems for continuous state branching processes; phase transition of particle systems and approximation of patterns and runs in finite state Markov chain. We are expecting some new results that may have influence and push forward the research in related fields.

分枝过程是应用随机过程的一个重要分支。广义分枝过程是经典GW分枝过程的推广，其结构更接近物种繁衍的真实画面，有重要的理论意义和极广泛的应用背景。最近10多年以来，随机树成为概率论领域的热点话题,并和分枝过程的研究紧密结合。粒子系统是数学与其他学科交叉的重要研究领域，许多应用学科中导出的模型都具有非常类似的数学结构。 极限理论是概率论研究的重要课题，马氏链的渐近估计是概率论的前沿方向。 对于"广义分枝过程和粒子系统"，该方向有不少为学术界关注的问题尚待解决。我们主要致力于以下几方面的研究： 广义分枝过程的树结构和条件极限；树值过程的剪切与极限；连续状态分枝过程的条件极限，粒子系统的相变和极限性质；同时也将研究多状态马氏链中patterns和runs分布的估计。期望在某些问题上的研究取得突破。

四年来，围绕着带移民分枝过程、随机树、随机游动这三条主线，我们在分枝过程的极限性质、随机树的构造与极限、分枝随机游动的渐近性质等课题的研究方面取得了一系列成果。共发表（或接受发表）研究论文10篇，其中7篇为SCI论文；另有已完成并投稿论文7篇。本项目所取得的主要成果为：研究了一般的CMJ分枝过程的Levy构造；证明了下临界CMJ分枝过程未来代粒子数的极限定理；研究了带依赖移民临界分枝过程的中心极限定理；带移民上临界分枝过程的调和矩与大偏差；研究了一类处于stable分布吸引域的独立随机变量随机和的极限；由GW过程控制的负相依随机变量随机和的大偏差；讨论了连续型随机树的剪切，过程的转移概率，增长性质，极限行为以及上临界Levy树的刻画；证明了两类离散的树值马氏过程收敛到连续性随机树值马氏过程；研究了stable分枝随机游动的鞅收敛定理、最小位置的二阶极限等等。在学术交流方面，积极参与国际国内合作与交流，多次参加国际国内学术会议。人才培养方面，培养了一定数量的硕士生和博士生，四年间共培养毕业博士生1人，硕士生6人；新增在读博士生4人，在读硕士生11人。