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The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. (English) Zbl 0963.60086
Let $$X$$ be a continuous-state branching process (CSBP), i.e., a Markov process $$X$$ with values in $$R_+$$ enjoying the property: if $$X^x$$ and $$X^y$$ are two independent copies of $$X$$ started at $$x$$ and $$y$$, resp., then $$X^x + X^y$$ has the law of $$X$$ started at $$x+y$$. The paper is focused on the case where $$X$$ has the branching mechanism $$\psi$$ given by $$\psi(u) 3D u \log u$$ [cf. J. Neveu, “A continuous-state branching process in relation with the GREM model of spin glass theory” (Rapport interne no 267, Ecole Polytechnique, 1992)]. In the main Theorem 4 the connection is shown of this Neveu’s CSBP with the coalescent process introduced by E. Bolthausen and A.-S. Sznitman [Commun. Math. Phys. 197, No. 2, 247-276 (1998; Zbl 0927.60071)] as a continuous-time Markov process $$(\Gamma_s, s \geq 0)$$ with values in the set of all equivalence relations on $$N$$. There are two proofs supplied: one relies on the relation between coalescent processes and Ruelle’s probability cascades [cf. D. Ruelle, ibid. 108, 225-239 (1987; Zbl 0617.60100)], and the other is direct. There is also analyzed the genealogy of CSBPs via Bochner’s subordination.

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G51 Processes with independent increments; Lévy processes
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