本博士学位论文共分为五章.


我们在第一章叙述了两物种分枝过程的研究背景和主要结果的综述.


在第二章, 我们给出了两物种分枝过程的极限定理, 用累积半群收敛的方法证明了两物种离散时空分枝过程的重整化极限是两物种连续时空分枝过程, 并且把结论推广到了带移民的情形. 累积半群收敛的方法把离散时空分枝过程与连续时空分枝过程之间的关系体现得更为清晰. 证明过程中涉及到一些二维矩阵层面的放缩技巧, 对于其他的多维度随机过程的研究, 尤其是生成元未知的随机过程的极限定理证明, 具有参考价值.


在第三章, 我们研究了两物种连续时空分枝过程的性质, 包括函数矩性质和局部最大跳的分布. 我们证明了, 两物种连续时空分枝过程函数矩的存在, 仅与过程的初分布和分枝机制中泊松随机测度的强度有关, 与漂移项和波动项没有关系. 过程的局部最大跳的分布由过程的初值与分枝机制共同决定. 两物种分枝过程的研究难点在于两物种之间的交互影响和数量上的不完全同步. 本文采取矩阵指数函数的工具和分区放缩的技巧来克服这一困难.


在第四章, 我们建立了莱维随机环境下的两物种连续时空分枝过程模型, 给出了过程的转移半群和随机积分方程构造. 这一结果填补了传统的两物种连续时空分枝过程模型与新兴的莱维随机环境下单物种连续时空分枝过程模型之间的空白, 采取了与二者都不相同的方式证明了随机积分方程组解的存在唯一性, 并且给出了过程的转移半群, 进一步证明了转移半群的费勒(Feller)性质.


论文第五章研究了莱维随机环境下两物种连续时空分枝过程的函数矩性质和正积分泛函, 随后构造了带移民的莱维随机环境下两物种连续时空分枝过程, 并且给出了过程遍历性的充分必要条件. 在限定的条件下, 带移民的过程极限分布的存在性取决于移民机制中泊松随机测度的强度. 这个结论与之前的单物种模型中相应结论是相容的. 但是由于两物种模型的复杂性, 证明过程相比较单物种模型复杂了许多. 即便是在没有随机环境的传统带移民多物种分枝过程模型中, 遍历性的等价条件也尚未见到结果发表. 因此本文中莱维随机环境下带移民的两物种连续时空分枝过程的遍历性结果具有创新性, 所运用的方法对于更广泛的高维仿射过程的研究具有一定的参考价值.


本文的所有方法和结论可以推广到任意有限多物种的分枝过程.

The thesis consists of five chapters.


In the first chapter, we give some backgrounds on the study of the two-type branching processes and a summary of the main results.


In the second chapter, we present the limit theorem for two-type branching processes both with and without immigration. By a method of convergence of cumulant semigroups, we prove that the scaling limits of Galton-Watson processes (with immigration) are continuous-state branching processes (with immigration). The method using cumulant semigroups presents more clearly the relation between processes. The proof concludes some inequalities of matrices. And it is helpful to other studies on multi-dimensional stochastic processes, especially on processes with unknown generators.


In the third chapter, we provide some results on the moment properties and local maximal jumps of two-type continuous-state branching processes. We find that, the finiteness of moments of two-type continuous-state branching processes only relates to the initial distribution and the intensity of the Poisson random measure in the branching mechanism. The distribution of local maximal jumps is determined by both the initial value and the whole branching mechanism. The difficulty of the study on two-type continuous-state branching processes is that the population of the two types is not totally synchronize and they interact with each other. We use the tool of matrix exponential and separating zones to conquer the difficulties.


In the fourth chapter, we establish the model of two-type continuous-state branching processes in L\'{e}vy random environments. The stochastic equations and the transformation semigroups are provided. The results fill the gap between the model of traditional two-type continuous-state branching processes and model of continuous-state branching processes in L\'{e}vy random environments. We prove the existence and uniqueness of the strong solution to the stochastic equations in a way different from the models above. We also proved the Feller property of the semigroup.


The last chapter studies the moment properties and positive integrals of the two-type continuous-state branching processes in L\'{e}vy random environments.  In addition, immigration is considered into the model and the equivalent condition for the ergodicity of the processes with immigration is presented at the end of this chapter. Under certain conditions, the existence of limiting distribution depends on the intensity of the Poisson random measure in the immigration mechanism. This conclusion is in consistence with the results of single-type continuous-state branching processes in L\'{e}vy random environments. But the proof is much more complex here. Until our discovery of the results, the equivalent condition of ergodicity was an unsolved problem even in the model of traditional multi-type continuous-state branching processes. Thus our results are innovative and the methods we use can be applied to more general models of multi-dimensional affine processes.


All results and methods in this thesis can be applied to any finite-type branching processes.