The Entrance Boundary of the Multiplicative Coalescent
David Aldous, University of California, Berkeley
Vlada Limic, University of California, Berkeley
Abstract
The multiplicative coalescent $X(t)$ is a $l^2$-valued
Markov process representing coalescence of clusters of mass, where each
pair of clusters merges at rate proportional to product of masses. From
random graph asymptotics it is known (Aldous (1997)) that there exists
a {it standard} version of this process starting with infinitesimally
small clusters at time $- infty$.
In this paper, stochastic calculus techniques are used to describe all
versions $(X(t);- infty < t < infty)$ of the multiplicative coalescent.
Roughly, an extreme version is specified by translation and scale parameters,
and a vector $c in l^3$ of relative sizes of large clusters at time $-
infty$. Such a version may be characterized in three ways: via its $t
to - infty$ behavior, via a representation of the marginal distribution
$X(t)$ in terms of excursion-lengths of a L{'e}vy-type process, or via
a weak limit of processes derived from the standard version via a ``coloring"
construction.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.