Coalescing systems of non-Brownian particles

Steven N. Evans, Ben Morris, Arnab Sen

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)307-342
Number of pages36
JournalProbability Theory and Related Fields
Volume156
Issue number1-2
DOIs
StatePublished - Jun 2013

Keywords

  • Brownian web
  • Coalescing particle system
  • Fractal
  • Hitting time
  • Stepping stone model

Fingerprint Dive into the research topics of 'Coalescing systems of non-Brownian particles'. Together they form a unique fingerprint.

Cite this