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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 densityis obtained when either the hitting time or the starting site $B_0$ becomeslarge. Our results entail that if Brownian motion is started at ${\bf x}$ andconditioned to hit $U(a)$ at time $t$ for the first time, the distribution ofthe 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; ineach case we provide a precise asymptotic estimate of the density. In the casewhen $|{\bf x}|/t$ tends to a positive constant we show the convergence of thedensity and derive an analytic expression of the limit density.

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