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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
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#### Convergence of Planar Domains and of Harmonic Measure Distribution Functions

Barton, Ariel 01 December 2003 (has links)
Consider a region Ω in the plane and a point z0 in Ω. If a particle which travels randomly, by Brownian motion, is released from z0, then it will eventually cross the boundary of Ω somewhere. We define the harmonic measure distribution function, or h-function hΩ, in the following way. For each r > 0, let hΩ(r) be the probability that the point on the boundary where the particle first exits the region is at a distance at most r from z0. We would like to know, given a function f, whether there is some region Ω such that f is the h-function of that region. We investigate this question using convergence properties of domains and of h-functions. We show that any well behaved sequence of regions must have a convergent subsequence. This, together with previous results, implies that any function f that can be written as the limit of the h-functions hΩn of a sufficiently well behaved sequence{Ωn}ofregionsis the h-function of some region. We also make some progress towards finding sequences {Ωn} of regions whose h-functions converge to some predetermined function f, and which are sufficiently well behaved for our results to apply. Thus, we make some progress towards showing that certain functions f are in fact the h- function of some region.

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