Brown运动的逗留时与首中时

设为中的标准Brown运动，对0＜α，记本文求出了X在首中球面之前逗留在Bα内的时间的Laplace变换，在首中之前逗留在Bαb内的时间的Laplace变换以及在首中之前逗留在Bαb内的时间的Laplace变换．作为推论，求出了X关于球面首中时的Laplace变换，逗留在球内总的时间的Laplace变换及逗留在球壳内的总的时间的LaPlace变换．

Sojourn Times and Hitting Times of Brownian Motion

Let X = (Xt)t be a d-dimensional Brownian motion in space Rd(d3), for0 < α, define Bα = {x: x∈Rd,|x| } and Bαb = {x: x ∈Rd,α}.In this paper, we compute the Laplace transforms of the sojourn time of Bα by Xbefore it hitting sphere Bb, the sojourn time of Bαb by X before it hitting Bα andthe sojourn time of Bαb by X before it hitting Bc, respectively. As corollaries, weobtain the Laplace transforms of the hitting time of a sphere by X, the total time Xspends in a bail and the total time X spends in a spherical shell, respectively.