On a Property of Harmonic Measure on Simply Connected Domains

Abstract

Let D ⊂ C be a domain with 0 ∈ D. For R > 0, letωD (R) denote the harmonic measure of D ∩ {|z| = R} at 0 with respect to the domain D ∩ {|z| < R} and ωD (R) denote the harmonic measure of ∂D ∩ {|z| ≥ R} at 0 with respect to D. The behavior of the functions ωD andωD near ∞ determines (in some sense) how large D is. However, it is not known whether the functions ωD andωD always have the same behavior when R tends to ∞. Obviously, ωD (R) ≤ωD (R) for every R > 0. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant C such that for all simply connected domains D with 0 ∈ D and all R > 0,In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of D, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

“…If is a discrete distribution, then Gross' construction yields a comb domain. Other examples include [26], in which a similar domain was used in order to construct a stopping time related to the winding of Brownian motion, and [17,18,19], in which similar domains were used as counterexamples to several conjectures concerning harmonic measure posed in [2,3]. Note that in Gross' paper in particular (see also [5,6,24]) the moments of the exit time are of importance, and yet it is not simple to show that they are finite for a given comb-like domain.…”

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

“…If µ is a discrete distribution, then Gross' construction yields a comb domain. Other examples include [26], in which a similar domain was used in order to construct a stopping time related to the winding of Brownian motion, and [17,19,18], in which similar domains were used as counterexamples to several conjectures concerning harmonic measure posed in [2,3]. Note that in Gross' paper in particular (see also [5,6,24]) the moments of the exit time are of importance, and yet it is not simple to show that they are finite for a given comb-like domain.…”

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

“…The main purpose of this section is to examine a recent result of Karafyllia [13] on the harmonic measure of starlike domains. We will show how the result can be extended, and will give a proof using Brownian motion.…”

“…without mentioning Brownian motion. This is the definition used by Karafyllia in [13], but the two definitions are equivalent.…”

“…An open set D on C is starlike (with respect to z 0 if there exists some The following result was proved analytically for starlike domains by Karafyllia [13], in order to answer a question posed by Betsakos in [4].…”