A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion

TL;DR

This paper explores the relationship between the geometry of planar domains and the exit times of Brownian motion, providing probabilistic proofs of conformal invariance and harmonic measure properties, along with examples and identities.

Contribution

It offers new probabilistic proofs of conformal invariance and harmonic measure results, and presents novel examples linking domain geometry with Brownian exit times.

Findings

Proof of conformal invariance of moduli using Brownian motion

Probabilistic proofs of harmonic measure results on starlike domains

Examples linking domain size to moments of Brownian exit time

Abstract

We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.