Colloquium TalkApril 7, 2014
Kevin Gerstle from the University of Iowa
Brownian Motion and HMD-Functions
3:30pm, Wednesday (4/9/2014)
Models of random motion can be used in all scientific disciplines to
study phenomena ranging from the rise and fall of stock prices to the
motion of pollen particles through water. In particular, models of
Brownian motion can be used to describe the behavior of random,
continuous motion through any number of dimensions.
By letting these randomly moving particles, ie: Brownian particles,
travel through a two-dimensional domain, we can learn valuable
information about the shape and structure of the domain by determining
how far from their starting points Brownian particles will first hit the
domain’s boundary. In doing so, we will construct the “harmonic measure
distribution functions” (HMD-functions) for these domains and discuss
just what information these functions encode. For example: are the
domains bounded? Do they have any corners or cusps? Do they have any
holes in them? Are their boundaries connected?
We will examine the HMD-functions of several different types of domains
using both geometrical and computational methods and then look in
particular at a class of circle domains for which the HMD-functions are
of particular interest. In doing so, we will use concepts from analysis,
geometry, probability, and computational theory in order to define these
functions and to explain their significance.