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Branching processes in Lévy processes: The exploration process. (English) Zbl 0948.60071
The fact that Brownian excursions can be used to code the genealogy of continuous-state branching processes with quadratic branching mechanism, underlies the construction of the Brownian snake. The latter is a path-valued process which is deeply connected to superprocesses with quadratic branching mechanism; see the recent monograph by the first author [“Spatial branching processes, random snakes, and partial differential equations” (1999; Zbl 0938.60003)]. The paper under review is motivated by the problem of extending the preceding construction to superprocesses with a general branching mechanism \(\psi\), using an analogue of the Brownian snake. In this direction, the authors consider a Lévy process \(Y\) with only positive jumps that has Laplace exponent \(\psi\), and introduce the so-called height process \(H\) as a local time related to \(Y\) by the fluctuation theory. They show that \(H\) provides a natural coding (analogous to the Brownian excursion in the quadratic case), by establishing that the occupation density process of \(H\) is a continuous state branching process with branching mechanism \(\psi\), which can be viewed as an extension of the Ray-Knight theorem. It is interesting to recall that a different connection between \(Y\) and the continuous state branching process with mechanism \(\psi\) has been pointed out by J. Lamperti [Bull. Am. Math. Soc. 73, 382-386 (1967; Zbl 0173.20103)].
Reviewer: J.Bertoin (Paris)

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