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Random walks, Brownian motion and branching processes. (Marches aléatoires, mouvement brownien et processus de branchement.) (French) Zbl 0741.60078

Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 258-274 (1989).
[For the entire collection see Zbl 0722.00030.]
The interesting links between the real Brownian motion and random walks on the one hand, and branching processes on the other hand, were observed first some twenty years ago, in connection with the Ray-Knight’s theorems on the Brownian local times. This paper pursue the study of these links, stressing on the notion of tree associated with branching processes.
The key result is a pathwise construction of the binary branching process (where the lifetime of a particle has an exponential distribution with parameter 1/2) from the inhomogeneous random walk \[ Z_ n:=\sum_{i=1}^ n(-1)^{i+1}Y_ i, \qquad i=1,\dots,n, \] where the \(Y_ i\)’s are i.i.d. standard exponential r.v.’s. This result is then applied to recover a recent theorem due to J. Pitman and J. Neveu [ibid., 239-247, 248-257 (1989; see below in this volume)] on the number of excursions of a Brownian motion above a level \(x\) and with height larger than \(h\). The author also describes the law under the Itô measure of the Brownian excursions of height \(>1\), of the number of excursions above the level \(x\) which reach the level 1, as an inhomogeneous branching process. Recently, the author pushed further these relations between Brownian exursions and branching processes, to provide a pathwise construction of the so-called ‘super Brownian motion’.
Reviewer: J.Bertoin (Paris)

MSC:

60J65 Brownian motion
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

Citations:

Zbl 0722.00030
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