In his 1979 Ph.D. thesis (Mathematics, UW-Madison) and in a subsequent 1981 uncompleted manuscript, Richard Arratia described a coalescing Brownian flow. Though his results were never published, his work is often cited, and connections with genetically motivated particle models have been observed in recent work (Donnelly et al., 2000; Evans and Zhou, 2004; Zhou, 2005) and, remarkably, an entirely new mathematical construction of the flow was given top billing in the October, 2004 issue of the Annals of Probability (Fontes et al.).
Arratia flow "lives" on the real line and evolves over time.
Intuitively, at every point in space and time, a bouncing baby
Brownian motion is born, and whenever two motions come together, they coalesce into a single motion. Technically, though, rigorously defining Arratia's uncountable system is difficult, and both Arratia and Fontes et al. use the usual sort of mathematical trickery where the flow is shown to exist without giving any real insight into what, precisely, it *is*.
In this talk, I will give a simple, straightforward, and concrete
construction of the flow. Unlike Arratia's original construction,
which considers only finite systems before invoking the Kolmogorov extension theorem or Fontes et al.'s which "completes" a countable object in an abstract metric space of equivalence classes of sets of particle paths (!?), the present approach should be accessible to those who would cross to the other side of the street if they ever saw a stochastic flow oozing up the sidewalk. |
