本学位论文分成两部分.第一部分, 本文先构造一类带移民的分枝随机游动, 直观来讲, 模型中的移民是嫁接过来的. 首先, 证明了由该过程构造的一个下鞅的收敛性, 给出了它极限存在的充要条件. 其次, 研究该过程中每一代粒子的空间位置的极限行为, 也称为中心极限定理. 一方面, 在临界和下临界时, 移民的引入, 使得过程非灭绝, 此时本文证明了过程在依分布收敛意义下的中心极限定理, 填补了该类结果在临界和下临界时的空缺; 另一方面, 在上临界带移民的情况下, 本文分别在依概率和平方矩的收敛下证明了中心极限定理. 为了研究上临界带移民的分枝随机游动, 本文还证明了带移民分枝过程的一些结果, 包括第n代粒子的最年轻的祖先的极限性质等, 这本身也是带移民分枝过程的重要性质.最后, 证明了带移民分枝随机游动第n代粒子最左位置构成的过程的大数定律, 这里的结果依赖于移民的数量, 当移民达到一定规模时, 移民的作用明显, 得到的大数定律的速率不同于没有移民的情形.第二部分, 本文研究带随机移民的分枝布朗运动的极限性质. 分枝布朗运动是分枝随机游动的连续版本, 其中粒子的运动是布朗运动. 该部分中, 假定移民是由另外一个分枝布朗运动决定的, 引入随机移民. 随机移民本质上是一种随机环境, 它的引入使得我们可以在两种概率---annealed 和quenched 概率下分别研究过程的性质. 第一, 本文证明了当空间运动的维数d大于等于3 时, 过程在annealed概率下的中心极限定理. 该结果依赖于空间维数, 在临界维数(d=4) 时, 移民的方式和粒子本身分枝游动的方式都对结果有贡献; 高维时, 粒子本身分枝游动的方式占主导地位; 低维时, 移民的方式占主导地位. 第二, 本文证明了当d大于等于4 时, 过程在quenched概率下的中心极限定理. 在临界维数(d=4) 时, quenched 概率和annealed概率下, 过程的波动有相同的阶, 但它们不是等价的.第三, 中心极限定理告诉我们过程会在它的均值附近波动, 在此基础上, 本文进一步考察过程偏离其中心区域这一小概率事件的衰减速度, 证明了d大于等于5时过程的大偏差原理.

This thesis consists of two parts.In the first part, we construct a branching random walk with immigration. In our model, the immigrants sits on some branches in the system.Firstly, we obtain the limit of a submartingale relative to this model,we give the necessary and sufficient condition under which the submartingale has a proper limit. Secondly, we consider the central limit theorem of our model. On the one hand, in the critical and subcritical cases, because of the immigrants, the model is non-extinct. And we prove the central limit theorems in the sense of convergence in distribution, which fill the gap of the central limit theorem. On the other hand, we get the central limit theorems in square mean and in probability in the supercritical case.To get the above results, we need to prove some results of the branching process with immigration at first, for example, the asymptotic behavior of the coalescence of the branching process with immigration, which is also important to the branching process with immigration itself. Thirdly, we prove the law of large number of the leftmost positions of the model. It reveals that when the number of the immigrants is large enough, the immigrants play important role, which makes the results different to those of the process without immigration.In the second part, we study the limit behaviors of branching Brownian motion with random immigration.Branching Brownian motion is the continuous time version of the branching random walk with particles making Brownian motion. In this part, the immigration is determined by another branching Brownian motion, which is the random immigration. In fact, the random immigration is a kind of random environment. Because of it, we can study the process under two kinds of laws--the $annealed$ law and the $quenched$ law. Firstly, we get the central limit theorem of the process when the underlying dimension $d\geq 3$ under the $annealed$ law.The result depends on $d$, when $d=4$, the critical dimension, the way of the immigration and the behaviors of the particles in the system both make contributions to the limit; when $d\ge 5$, the way of the behaviors of the particles in the system plays the dominant role; when $d=3$, the way of the immigration plays the dominant role.Secondly, we prove the central limit theorem when $d\ge4$ under the $quenched$ law. When $d=4$, the fluctuation of the process under the two kinds of laws are the same, but they are not equal.Thirdly, having the result of the central limit theorem, we can consider theasymptotic rate of the probability of the process deviating from the neighborhood of the centers. We prove the large deviation principle when $d\ge5$.