*Analytical and Numerical Results for Brownian Motion Through a Material*

**Carey Caginalp**

University of Pittsburgh

Department of Mathematics

`cac71@pitt.edu`

**Abstract**: A particle moves with Brownian motion in a unit disc
with reflection from the boundaries except for a portion (called ``window'' or
``gate'') in which it is absorbed. This can simulate, for example, impurities
diffusing randomly through materials. It can also illustrate two chemical
molecules A and B, where A is far more massive and hence effectively
stationary, and where B drifts around until it strikes B, causing a chemical
reaction to occur. As a biological application, an ion may drift around a
cell, bouncing off the outside membrane, unless it hits a small pore and
escapes, causing an electric imbalance in the cell. The results given include
a closed formula for the mean first hitting time is given for a gate of any
size. Also given is the probability density of the location where a particle
hits if initially the particle is at the center or uniformly distributed.
Numerical simulations of the stochastic process with finite step size and
sufficient amount of sample paths are compared with the exact solution to the
Brownian motion (the limit of zero step size), providing an empirical formula
for the difference. Histograms of first hitting times will also be
given.

This work is in collaboration with Professor Xinfu Chen. The first paper has appeared recently in Comptes Rendus Matematique.