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1998 The Entrance Boundary of the Multiplicative Coalescent
David Aldous, Vlada Limic
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Electron. J. Probab. 3: 1-59 (1998). DOI: 10.1214/EJP.v3-25


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 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 Lévy-type process, or via a weak limit of processes derived from the standard version via a "coloring" construction.


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David Aldous. Vlada Limic. "The Entrance Boundary of the Multiplicative Coalescent." Electron. J. Probab. 3 1 - 59, 1998.


Accepted: 19 January 1998; Published: 1998
First available in Project Euclid: 29 January 2016

zbMATH: 0889.60080
MathSciNet: MR1491528
Digital Object Identifier: 10.1214/EJP.v3-25

Primary: 60J50
Secondary: 60J75

Keywords: entrance boundary , excursion , Lévy process , Markov process , random graph , stochastic coalescent , weak convergence

Vol.3 • 1998
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