布朗运动跑出时的光滑性

Smoothness of exit time in Brownian motion

对布朗运动停时的光滑性进行了研究 .运用函数的正交分解方法 ,证明了只需当分数次索伯列夫空间的可积性指标与可微性指标的乘积小于 1这一条件满足时 ,布朗运动从某个开集的跑出时属于这个分数次索伯列夫空间 ,解决了以往证明过程中可积性指标受状态空间维数约束的问题 .

The smoothness of stopping times in Brownian motion is studied. It has been proved by the orthogonal expansion of the function that the exit time of Brownian motion out of an open set belongs to some fractional Sobolev space, provided that the product of this Sobolev space's differentiability and integrability indexes is less than unity. This result has resolved the problem of the constraint on the integrability index by the dimension of the state space.