Ariel Barton

Harvey Mudd College Mathematics 2003

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Thesis Advisor: Prof. Lesley Ward
Second Reader: Prof. Henry Krieger
E-Mail: abarton@thuban.ac.hmc.edu
Thesis: Conditions on harmonic measure distribution functions of planar domains

Conditions on harmonic measure distribution functions of planar domains

Consider a region D in the plane and a point z in D. If a particle which travels randomly, by Brownian motion, is released from z, then it will eventually cross the boundary of D somewhere. We define the harmonic measure distribution function, or h-function, in the following way. For each r>0, let h(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 z. We would like to know, given a function f, whether there is some region D 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 of a sufficiently well-behaved sequence {D_n} of regions is the h-function of some region.

We also make some progress towards finding sequences {D_n} 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.