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 ?

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## Alexander Shamov said,

November 25, 2010 at 4:18 pm

– So what if we do that on a segment?

We’ll get nothing more than the time of Brownian motion hitting a point, since the whole lot of particles coalesce once the leftmost and the rightmost ones do. 🙂

## Alekk said,

November 25, 2010 at 4:29 pm

indeed, that was a (bad) joke 🙂

## Benjamin o. said,

June 9, 2011 at 6:54 pm

Pls, I need your help in practical Biology(NECO) Exam that coming up on 10/6/2011