Brownian hitting distributions in space-time of bounded sets and the expected volume of Wiener sausage for Brownian bridges

Abstract

The space-time distribution, Q A (x, dtdξ) say, of Brownian hitting of a bounded Borel set A of R d is studied. We derive the asymptotic form of the leading term of the timederivative Q A (x, dtdξ)/dt for each d = 2, 3, . . ., valid uniformly with respect to the starting point x of the Brownian motion, which result significantly extends the classical ones for Q A (x, dtdξ) itself by Hunt (d = 2), Joffe and Spitzer (d ≥ 3). The results obtained are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joining the origin to a distant point.