Coalescing Brownian flow

A coalescing Brownian flow describes particles that follow independent one-dimensional Brownian motions until they meet, after which they move together. For any finite collection of deterministic space-time starting points, this rule specifies a consistent interacting system. Extending it to trajectories from all space-time points requires a suitable state space: separate one-particle distributions do not record which histories merge.^[ref]

Two particles give an illuminating calculation. Start them at $0$ and $a>0$, with the normalization $\mathrm{Var}(B_t)=t$, and let $T$ be their first meeting time. Before meeting, their separation has the first-hit law of $a+\sqrt{2}{W}_t$, because the difference of independent Brownian increments has twice their variance. The Brownian reflection principle giveswhere $\Phi$ is the standard normal distribution function. For $a=1$ and $t=1$, the meeting probability is about $0.4795$. After a meeting the two positions are identical, although each position separately still has the appropriate Brownian marginal law. Continuing to drive both particles independently would describe a different model.

The flow connects microscopic merging trajectories with macroscopic random geometry. Nathanaël Berestycki, Christophe Garban and Arnab Sen developed a state-space description in which rescaled coalescing random walks converge under a finite-variance assumption. Their result improves the moment requirements needed by earlier descriptions; it does not identify the origin of the subject with that later theorem.^[ref]