Coalescing systems of non-Brownian particles

TL;DR

This paper explores the conditions under which systems of coalescing particles, including those with discontinuous paths or on complex spaces, coalesce into finite sets, extending classical results beyond Brownian motion on the real line.

Contribution

It provides a general criterion for coalescence in systems with non-standard state spaces or discontinuous paths, and demonstrates almost sure instantaneous coalescence in specific fractal and stable process cases.

Findings

Coalescence occurs almost surely in systems on the Sierpinski gasket.

Stable processes with index > 1 coalesce instantaneously.

The criterion applies to a broad class of non-Brownian systems.

Abstract

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one.