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

The space-time distribution, $Q_A(x,dtd\xi )$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtd\xi)/dt$foreach$d =2, 3, …$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x,dtd\xi )$ itself by Hunt ($d=2$), Joffe and Spitzer ($d=3,4,\mathrm{.}\mathrm{.}\mathrm{.}$). The results 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.