| Abstract: | We compute the joint distribution of the site and the time at which a d-dimensional standard Brownian motion ((B˙t)) hits the surface of the ball ((U(a) ={—x—<a})) for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site ((B˙0)) becomes large. Our results entail that if Brownian motion is started at ((x)) and conditioned to hit ((U(a))), at time t, the distribution of the hitting site approaches the uniform distribution or the point mass at ((ax/—x—)) according as ((—x—/t)) tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when ((—x—/t)) tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density. |
