Brownian Exit Distribution of a Ball

Let B be a ball in Rd and X = {Xt,t ≥ 0} be the standard Brownian motion in Rd. Define τB = inf{t > 0: Xt ∉B}, the first exit time of X from the ball. We compute explicitly the transition density function of the killed Brownian motion Xo = {Xt, t < τB} and the joint distribution of (τB,X(τB)}. A result of Wendel [5] is deduced as a simple consequence of the explicit joint density function.