Today, James norris gave a talk related to Diffusion-limited aggregation processes and mentioned, in passing, the following amusing fact: put equidistant Brownian particles on the circle with unit circumference and let them evolved. When two of them collide they get stuck to each other and continue together afterwards: after a certain amount of time , only one particle remains. Perhaps surprisingly, this is extremely easy to obtain the first few properties of . For example, .
To see that, define the distance between and (modulo ) so that for . Notice then (It\^o’s formula) that
is a (local) martingale that starts from . Also, at time , exactly of the distances are equal to while one of them is equal to : this is why . The end is clear: apply the optional sampling theorem (to be rigorous, take not too big, or do some kind of truncations to be sure that the optional sampling theoem applies) to conclude that
This gives for example . I just find it cute!
So what if we do that on a segment ?