On the expected exit time of planar Brownian motion from simply connected domains

TL;DR

This paper investigates the expected exit time of planar Brownian motion from simply connected domains, providing new formulas, bounds, and applications including classical results and non-simply connected cases.

Contribution

It introduces a simple formula for expected exit times, applies it to various domains, and extends the analysis to non-simply connected regions, connecting complex analysis with probabilistic methods.

Findings

Derived a formula for expected exit time in simply connected domains.

Applied the formula to specific shapes like cardioids, polygons, and strips.

Connected probabilistic exit times with classical mathematical constants.

Abstract

This paper presents some results on the expected exit time of Brownian motion from simply connected domains in $\CC$. We indicate a way in which Brownian motion sees the identity function and the Koebe function as the smallest and largest analytic functions, respectively, in the Schlicht class. We also give a sharpening of a result of McConnell's concerning the moments of exit times of Schlicht domains. We then show how a simple formula for expected exit time can be applied in a series of examples. Included in the examples given are the expected exit times from given points of a cardioid and regular $m$-gon, as well as bounds on the expected exit time of an infinite wedge. We also calculate the expected exit time of an infinite strip, and in the process obtain a probabilistic derivation of Euler's result that $\zeta(2)=\sum_{n=1}^\ff \frac{1}{n^2}= \frac{\pi^2}{6}$. We conclude by showing how the formula can be applied to some domains which are not simply connected.