# Ariel Barton

Harvey Mudd College Mathematics 2003

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.