Diffusion with “collisions” between particles

Abstract

First consider two particles diffusing on the same line, with positions at time t given by y₁(t) and y₂(t) respectively. We suppose that they cannot pass one another, so that if initially y₁(0) < y₂(0), then . As long as the two particles are not in contact, we suppose that each moves, independently of the other, according to the Wiener process. We must also prescribe what happens when collisions occur. There seems to be no unique natural way to do this, and we shall adopt a method recommended by its mathematical convenience. However, we shall see that there are two possible motivations for the prescription, one in terms of random walks, the other in terms of a more mechanical model.