#### Graduation Year

2003

#### Document Type

Open Access Senior Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### Reader 1

Lesley Ward

#### Reader 2

Henry Krieger

#### Abstract

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.

#### Recommended Citation

Barton, Ariel, "Convergence of Planar Domains and of Harmonic Measure Distribution Functions" (2003). *HMC Senior Theses*. 159.

https://scholarship.claremont.edu/hmc_theses/159

*Thesis Proposal*