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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.

Vitalii Konarovskyi is a mathematician working in probability theory working on stochastic analysis and interacting particle systems. In his paper on coalescing heavy diffusion particles, he constructed a modified Arratia flow in which particles carry conserved mass and their diffusion rate is inversely proportional to that mass, starting from uniformly distributed mass on an interval.^[ref]

The mechanism is easy to picture: after particles merge, the heavier cluster diffuses more slowly. A rigorous construction must accommodate a continuum of initial particles, rather than only a finite collection of independent paths. His work develops a stochastic integral for the resulting flow and describes total local time through an occupation measure. His authored notes on the large deviation principle also explain how unlikely fluctuations acquire quantitative path costs.^[ref]