OCCUPATION TIMES OF BALLS BY BROWNIAN MOTION WITH DRIFT

Let X = {Xt,t 0} be a d-dimensional (d 2) standard Brownian motion with drift c started at a fixed x, and BR = ｛x ∈ Rd : ｜x｜ ＜ R｝, the ball centered at 0 with radius R. Consider the distributions of TR(t) and TR(∞), where TR(t) denotes the time spent by X in BR up to time t and TR(∞) the total time of X spent in BR. Explicit formulas for the Laplace transform of TR(∞) and the double Laplace transform of TR(t) are obtained.