Brownian运动首次离开指定边界的概率表达式

The probability expressions of first exit for Brownian motions at prescribed boundary parts of regions

Brownian运动在随机过程与Schramm-Loewner演变(SLE)中扮演着非常重要的角色。首先,利用Brownian运动的性质导出了Brownian运动首次离开指定区域边界的概率表达式,并且通过数值模拟进一步验证了所得到结果的正确性。其次,讨论单连通域内调和测度与Brownian运动首离指定边界概率之间的关系,通过后者的结果给出了前者的具体表达式,同时得到一些相关的结果。

Brownian motion plays a very important role in both stochastic process and Schramm Loewner Evolution （SLE）.Firstly,by using the properties of Brownian motions,it is derived that the probability expressions of Brownian motion exiting first at prescribed boundary parts of regions.The correctness of the obtained results is verified by numerical simulations.Secondly,the relationship between harmonic meas-ure and probability of Brownian motion exiting first a simply connected domain at the specified boundary is discussed.The probability formulas of the former are derived by using the results of the latter.In addi-tion,some other related results are obtained.