The Annals of Probability

Brownian Excursions, Trees and Measure-Valued Branching Processes

Jean-Francois Le Gall

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Abstract

We propose a trajectorial construction of a class of measure-valued Markov processes, called superprocesses or measure-valued branching processes, which have been studied extensively in the last few years. These processes were originally defined as weak limits of systems of branching particles. The basic idea of our construction is to use the branching structure of excursions of a linear Brownian motion to model the branching mechanism of the superprocess. Without any additional effort, our approach leads to the so-called historical process, which contains more information than the superprocess in the sense that it keeps track of the individual paths followed by the particles. We emphasize the relationship between the properties of the historical process and the corresponding results of excursion theory. We also give a description of the support of the superprocess at a fixed time, using a simple tree model. Finally, we use our construction to recover certain pathwise properties recently obtained by Perkins.

Article information

Source
Ann. Probab., Volume 19, Number 4 (1991), 1399-1439.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990218

Digital Object Identifier
doi:10.1214/aop/1176990218

Mathematical Reviews number (MathSciNet)
MR1127710

Zentralblatt MATH identifier
0753.60078

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J65: Brownian motion [See also 58J65] 60G57: Random measures 60G17: Sample path properties 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65]

Keywords
Superprocess branching process Brownian excursions local time tree Ito excursion theory historical process sample path property random measure Palm measure

Citation

Gall, Jean-Francois Le. Brownian Excursions, Trees and Measure-Valued Branching Processes. Ann. Probab. 19 (1991), no. 4, 1399--1439. doi:10.1214/aop/1176990218. https://projecteuclid.org/euclid.aop/1176990218


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