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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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