# Colloquium Talk

*April 7, 2014*

**Kevin Gerstle** from the University of Iowa

**Brownian Motion and HMD-Functions**

3:30pm, Wednesday (4/9/2014)

VH1224

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.