Electronic Journal of Probability

Branching processes seen from their extinction time via path decompositions of reflected Lévy processes

Miraine Dávila Felipe and Amaury Lambert

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We consider a spectrally positive Lévy process $X$ that does not drift to $+\infty $, viewed as coding for the genealogical structure of a (sub)critical branching process, in the sense of a contour or exploration process [34, 29]. We denote by $I$ the past infimum process defined for each $t\geq 0$ by $I_t:= \inf _{[0,t]} X$ and we let $\gamma $ be the unique time at which the excursion of the reflected process $X-I$ away from 0 attains its supremum. We prove that the pre-$\gamma $ and the post-$\gamma $ subpaths of this excursion are invariant under space-time reversal, which has several other consequences in terms of duality for excursions of Lévy processes. It implies in particular that the local time process of this excursion is also invariant when seen backward from its height. As a corollary, we obtain that some (sub)critical branching processes such as the binary, homogeneous (sub)critical Crump-Mode-Jagers (CMJ) processes and the excursion away from 0 of the critical Feller diffusion, which is the width process of the continuum random tree, are invariant under time reversal from their extinction time.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 98, 30 pp.

Received: 24 August 2017
Accepted: 4 September 2018
First available in Project Euclid: 25 September 2018

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Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G17: Sample path properties 60J55: Local time and additive functionals 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

space-time reversal excursion measure duality Ray-Knight theorem Williams decomposition Lévy process conditioned to stay positive Esty transform

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Dávila Felipe, Miraine; Lambert, Amaury. Branching processes seen from their extinction time via path decompositions of reflected Lévy processes. Electron. J. Probab. 23 (2018), paper no. 98, 30 pp. doi:10.1214/18-EJP221. https://projecteuclid.org/euclid.ejp/1537841130

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