= 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.<Coalescing Brownian flows: A new approach by Nathanaël Berestycki and others>[Section 1.1, electronic reprint pages 1–2: finite-particle interpretation and difficulty of uncountably many starting points]{ref}
Two particles give an illuminating calculation. Start them at $0$ and $a>0$, with the normalization $\operatorname{Var}(B_t)=t$, and let $T$ be their first meeting time. Before meeting, their separation has the first-hit law of $a+\sqrt2 W_t$, because the difference of independent <Brownian increments> has twice their <variance>. The <Brownian reflection principle> gives
$$
\mathbb P(T\le t)=2\left[1-\Phi\left(\frac{a}{\sqrt{2t}}\right)\right]
=\operatorname{erfc}\left(\frac{a}{2\sqrt t}\right),\qquad t>0,
$$
where $\Phi$ 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.<Coalescing Brownian flows: A new approach by Nathanaël Berestycki and others>[Abstract and section 1.1: earlier constructions and finite-variance invariance principle]{ref}
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