小值概率理论是概率极限理论中的经典研究方向。变动维数随机过程理论又是近年来概率论中的新兴研究领域。结合上述两个概率方向的研究成果，本项目拟开展可变、随机维数下布朗运动出逃概率渐近估计的研究。具体内容包括：可变维数(即维数关于时间单调不减)下, 布朗运动出逃概率的渐近估计；随机维数(即维数是计数过程,例泊松过程)下, 布朗运动出逃概率的渐近估计。对于前者的研究主要利用国外相关学者、申请人及其团队关于常数维数下布朗运动出逃概率已有成果以及变动维数随机过程的相关理论；对于后者的研究主要借助于前者的研究成果以及计数过程(尤其是泊松过程)的相关理论、条件数学期望演算等基本理论。此项目的研究不仅是对常数维数下布朗运动出逃概率问题的深入与发展，同时其理论成果也是生物数学(例如种群控制)、金融保险(例如破产理论)等应用数学学科中多个问题的理论基础。

Small deviation probability is a classical research direction in the probability limit theory. Stochastic process theory with varying dimension is a new research direction in the probability theory. Based on these two research directions, in this project，we consider the escape probability of Brownian motion, it includes two subjects. First, the escape probability of Brownian motion with varying dimension in some domains; Second, the escape probability of Brownian motion with random dimension (such as Possion process). For the first one, it is mainly based on the works of many international scholars and applicant's team in the constant dimension and the results of stochastic process with varying dimension; For the second one, it is mainly based on the stochastic process (Possion process) theory and conditional expectation in advanced probability. The results of this project is not only an extension of the escape probability of Brownian motion with constant dimension, but also have broad applications in biomathematics and insurance mathematics.