本文考虑的是 Z 上的带有两个催化点的连续时间的分支随机游动模型, 催化点为原点和 x=1. 初始时刻在催化点原点有一个粒子, 粒子在催化点的停留时间服从参数为 1 的指数分布，随后粒子可能进行分裂也可能跳出催化点. 若粒子以一定概率向左跳到 x=-1, 随后粒子将在x=-1 左侧做简单随机游动; 若粒子以相同概率向右跳到催化点 x=1, 粒子在 x=1 的演化规则同催化点原点处; 若粒子以一定概率在催化点原点进行分裂产生随机个后代粒子, 后代母函数为 f(s), 产生的后代按照上述规则继续相互独立的运动. 本文证明了在临界状态下灭绝概率的渐近行为，同时建立了关于系统中粒子数的 Yaglom 条件极限。
技术上, 我们主要利用了带有四种粒子的 Bellman-Harris 过程, 证明了四种粒子的数目与模型中四个位置的粒子数目同分布, 进而可以借助熟悉的 Bellman-Harris 过程来解决问题. 通过本文的结论可以进一步得出, 模型中带有一个催化点还是带有两个催化点, 对系统中粒子灭绝概率的渐近行为影响并不大, 两种模型中的概率是同阶的.

In this paper, we consider a model of critical continuous time branching random walk with two catalytic points 0 and 1 on Z. The
population is initiated at time t=0 by a single particle. At the origin it spends an exponentially distribution time with parameter 1 and then either dies or jumps out of the catalytic point. If the particle jumps to the point -1 to the left with probability (1- alpha)/2, then the particle will performs a continuous time simple random walk on Z. If the particle jumps to the catalytic point 1 to the right with probability (1-alpha)/2, the movement rule of the particle at point 1 is the same as point 0. If the particle dies at the catalytic point 0 with probability alpha producing just before its death a random number of children in according with the offspring generating function f(s), the new particles behave independently and stochastically in the same way as the parent particle. In this article, we proved the asymptotic behavior of the probability of extinction, and we establish a Yaglom type conditional limit theorem for the number of particles in the system.
We mainly apply critical Bellman-Harris process with four types of particles in this paper and proved that the number of particles in the four positions in the model coincides in the distribution with the number of the four types of particles and then study with the tools of Bellman-Harris process. From the conclusions of this article, it can be further concluded that whether there is one catalytic or two catalytic points, it has little effect on the asymptotic behavior of extinction probability.