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Carnegie Mellon Center for Nonlinear Analysis
Analytical and Numerical Results for Brownian Motion Through a Material


Carey Caginalp
University of Pittsburgh
Department of Mathematics

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.