带形上分枝随机游动的极限定理

The prime concern of this program is to consider the branching random walk on a strip. Consider a strip consists of two lines, say, line a and b. Particles on line a generate a number of offsprings according to some branching mechanism and the particles do simple random walk. Especially particles on line a immigrate to line b with some positive probability. Particles on line b also branch and walk but will not immigrate to line a.. Naturally, in this program, we first consider the recurrence vs transience and also the limiting velocity of partilce system on line a and b. Since the particle system involves immigration, it’s different from the traditional branching random walk on lattice. But we can study the limiting behaviors of particle system on line a and b with the help of branching random walk on lattice.. The most interesting concern of this model is that it provides a realization of branching random walk with immigration (Hong-Sun 2011). The particle system on line b is indeed a "branching random walk with branching-random-walk immigration", that is, the immigration itself is also a branching random walk. Therefore, to study the model, we have to study the limiting behaviors of "branching random walk with branching-random-walk immigration". This is the main point and interest of this program.

本项目旨在研究带形上的分支随机游动。考虑最简单的由 a, b 两根线组成的带形， a 线上的粒子以一定的分枝机制产生后代，并作随机游动，在单位时间内粒子以一定的概率向b线上移民。b 线上的粒子也以一定的机制产生后代并做随机游动，但不向 a 线移民。. 本项目首先旨在研究清楚 a，b 两条线上粒子的常返和暂留性以及 a，b 两条线上的随机过程的运动速度，由于两条线中有移民，所以该模型和直线上的分枝随机游动有着本质的区别。但是可以借助于直线上分枝随机游动的性质研究该模型的性质。. 有趣的是，该模型给出了“带分枝随机游动移民的分枝随机游动游动”的一个实现，实质上，要研究清楚该模型的极限性质，必须证明“带分枝随机游动移民的分枝随机游动”的极限性质，这也是本项目最有意义且具有挑战性的工作，是本项目所要解决的核心问题。

项目执行期间，项目组证明了带型上分枝随机游动的相关极限定理，并研究了与之相关的多物种分枝过程和随机游动的极限性质。首先，项目研究了带型上不同层上分枝随机游动的性质，证明了a层上粒子的分枝与运动构成一个带移民（有粒子移出）的分枝随机游动，给出了a线上分枝随机游动常返暂留的判定条件，证明了大数定律并利用大偏差刻画了其准确的速度；刻画了b层上的粒子是一个多物种的分枝随机游动（年龄大于1的粒子不产生后代），并研究了b层上粒子的常返与暂留性。其次，项目刻画了临界（下临界）多物种分枝过程总人口的母函数，进而研究了整数轴上随机游动（有界跳幅）首中时的具体分布。再次，本项目以随机矩阵大偏差为工具，研究了一类随机环境中随机游动的次线性增长性质。最后，项目执行期间，申请人发表了关于随机环境中多物种带移民分枝过程总人口的尾概率估计的文章，这也是和本项目密切相关的。