Last Exit Time and Harmonic Measure for Brownian Motion in Rd

Abstract

The purpose of this paper is to establish a connection between the last exit time distribution and the so-called harmonic measure of the Brownian motion process in Rd . The main tool in handling this problem is the following Chung’s formula [1] : let D be a bounded domain in Rd , (X(t)) the Brownian motion process in Rd , and τD the exit time from D, i.e., τD = inf{t> 0; X(t) є DC} . For every bounded measurable function f on ∂D define HD as: $${{H}_{D}}f\left( x \right)={{E}^{X}}\left[ f\left( X\left( {{\tau }_{D}} \right) \right);{{\tau }_{D}}<\infty \right],x\in D$$ (*)Then, according to [1], a measure HD(x,.) defined on the boundary ∂D is called the harmonic measure of x with respect to D.