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

Aldous, David; Limic, Vlada

Electron. J. Probab. 3, Paper 3, 59 p. (1998).

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)

Keywords:

Markov process; entrance boundary; excursion; Lévy process; random graph; stochastic coalescent; weak convergence

Citations:

Zbl 0877.60010