Density of space-time distribution of Brownian first hitting of a disc and a ball

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) =\{|{\bf 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 ${\bf x}$ and conditioned to hit $U(a)$ at time $t$ for the first time, the distribution of the hitting site approaches the uniform distribution or the point mass at $a{\bf x}/|{\bf x}|$ according as $|{\bf x}|/t$ tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when $|{\bf x}|/t$ tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.