自Dawson与Fleischmann (1997) 首次引入测度值意义下的催化分枝过程以来，在这个方向有了许多工作. 本文的主要目的是考虑带移民的离散状态催化分枝模型, 我们称之为 催化CBI-过程. 此类模型可以定义为一类随机积分方程的强解.我们主要关心的是在不同的重整化(rescaling) 之下催化DBI-过程的极限定理问题. 对一列催化DBI-过程选取自然的scaling 考虑高密度极限, 我们得到一类新的具有催化作用的CBI-过程,此类过程不同于Dawson 与 Li(2006) 引入的催化 CBI-过程,但从高密度极限的角度看, 前者显得更为自然. 对一列催化DBI-过程考虑高密度波动极限，以及对一列新的催化CBI-过程考虑低密度波动极限, 我们所得到的极限过程皆为带非负跳的仿射马氏过程. 后者在数理金融中具有广泛的应用; 参见 Duffie 等 (2003). 在这里我们需要指出的是, 为了得到上述两类带跳的极限过程, 我们证明了相应的‘重整化概率母函数' 序列在一组简单条件下, 其极限函数具有Lévy-Khinchin 型表示. 该表示结果在证明上述离散模型的极限定理中发挥了重要的作用. 同时我们利用它也证明了一类二维 ~CBI-过程可由两物种带移民的G-W 过程经过自然的重整化依轨道弱收敛得到, 这推广了Li (2005)的部分结果.

Catalytic branching processes were introduced by Dawson and Fleischmann (1997) in the measure-valued setting and have been studied by many authors. The main objective of this work is to consider the proper formulation of the discrete state counterpart ofthe catalytic branching processes which we call catalytic discrete state branching processes with immigration (catalytic DBI-processes). The catalytic DBI-processes are defined as strong solutions of stochastic integral equations. We provide main theorems of these processes using different scalings. By considering the high density limits of those processes, we obtain a kind of catalytic CBI-processes which are slightly different from the models of Dawson and Li (2006). We also prove fluctuation limit theorems of the two classes of catalytic branching models, which lead to the same kind of affine processes in high density/low density fluctuations. Affine Markov processes have been used widely in mathematical finance; see, e.g., Duffie et al.(2003). Here we give some Lévy-Khinchin type representation results of the limits of the re-scaled bivariate probability generating functions, which play an important role in characterizing our limit processes. In addition, based on one of epresentation results, we also show that a class of two-dimensional CBI-processes can be obtained from the two-type GWI-processes by rescaling and passing to the limit, this generalizes part of limit theorems of Li (2005).