running head: space-time distribution of Brownian first hitting key words: harmonic measure for heat operator, Brownian hitting time, caloric measure, parabolic measure, space-time distribution AMS Subject classification (2010): Primary 60J65, Secondary 60J45, 60J60.
AbstractWe 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 for the first time, 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.