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.
| Identifer | oai:union.ndltd.org:CLAREMONT/oai:scholarship.claremont.edu:hmc_theses-1162 |
| Date | 01 December 2003 |
| Creators | Barton, Ariel |
| Publisher | Scholarship @ Claremont |
| Source Sets | Claremont Colleges |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | HMC Senior Theses |
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