Convergence properties of harmonic measure distributions for planar domains

We establish sufficient conditions under which the harmonic measure distribution functions h n of a sequence of domains D n converge pointwise to the distribution function h of the limiting domain D, at all points of continuity of h. In the case of a model example, we establish this convergence of t...

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Published in:Complex variables and elliptic equations 2008-10, Vol.53 (10), p.897-913
Main Authors: Snipes, Marie A., Ward, Lesley A.
Format: Article
Language:English
Subjects:
Brownian motion
Carathéodory convergence
Fréchet convergence
harmonic measure
harmonic measure distribution functions
planar domains
primary 30C85
secondary 30C20
step functions
weak convergence of measures
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description We establish sufficient conditions under which the harmonic measure distribution functions h n of a sequence of domains D n converge pointwise to the distribution function h of the limiting domain D, at all points of continuity of h. In the case of a model example, we establish this convergence of the distribution functions. Here, the value of the function h(r) gives the harmonic measure of the part of the boundary of the domain that lies within distance r of a fixed basepoint in the domain, thus relating the geometry of the domain to the behaviour of Brownian motion in the domain.
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subjects Brownian motion
Carathéodory convergence
Fréchet convergence
harmonic measure
harmonic measure distribution functions
planar domains
primary 30C85
secondary 30C20
step functions
weak convergence of measures
title Convergence properties of harmonic measure distributions for planar domains
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