## The entrance boundary of the multiplicative coalescent.(English)Zbl 0889.60080

Summary: The multiplicative coalescent $$X(t)$$ is an $$l^2$$-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 [D. Aldous, Ann. Probab. 25, No. 2, 812-854 (1997; Zbl 0877.60010)] that there exists a standard version of this process starting with infinitesimally small clusters at time $$-\infty$$. 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.

### MSC:

 60J50 Boundary theory for Markov processes 60J75 Jump processes (MSC2010)

Zbl 0877.60010
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