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Convergence of Planar Domains and of Harmonic Measure Distribution FunctionsBarton, 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 hfunction 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 hfunction of that region. We investigate this question using convergence properties of domains and of hfunctions. 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 hfunctions hΩn of a sufficiently well behaved sequence{Ωn}ofregionsis the hfunction of some region. We also make some progress towards finding sequences {Ωn} of regions whose hfunctions 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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