Arratia flow

An Arratia flow is a system of coalescing one-dimensional Brownian motions: particles starting at different positions move independently until they meet, then follow the same Brownian path forever. Starting from ordered positions preserves their order because paths do not cross before merging. The full construction makes these finite-particle rules consistent for a continuum of starting points.^[ref]

For two standard Brownian particles starting distance $a>0$ apart, their difference before meeting has variance $2t$. If $\tau$ is their meeting time, the Brownian reflection principle giveswhere $\Phi$ is the standard-normal distribution function. This tends to zero as $t\to \infty$, so they eventually merge almost surely, although the mean meeting time is infinite because the tail decays only as $t^{-1/2}$.

The same rule arises as a scaling limit of coalescing random walks. It models histories that can join but cannot later separate; reversing time does not simply produce independent nonbranching Brownian paths.