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

TL;DR

This paper investigates the space-time distribution of Brownian motion hitting bounded sets and derives asymptotic formulas, extending classical results, with applications to the expected volume of Wiener sausages for Brownian bridges.

Contribution

It provides new asymptotic formulas for the time-derivative of the hitting distribution, extending classical results to a broader context and applying them to Wiener sausage volume estimates.

Findings

Asymptotic form of the time-derivative of hitting distribution derived

Results extend classical findings by Hunt, Joffe, and Spitzer

Applied to asymptotic expected volume of Wiener sausage for Brownian bridges

Abstract

The space-time distribution, $Q_A(x,dt d\xi)$ 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 $Q_A(x, dtd\xi)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt d\xi)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). 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.