The genealogy of continuous-state branching processes with immigration
type: Item
instance of: scholarly article
title: The genealogy of continuous-state branching processes with immigration
zbMATH Open document ID: 1020.60074
author: Amaury Lambert
publication date: 2003-09-30T00:00:00Z
review text: J. F. Le Gall and Y. Le Jan have extended the genealogical structure of Galton-Watson processes to continuous state branching processes via a non-Markovian process called the height process. In this paper, continuous state branching processes with immigration are studied. The height process \(H\) is defined as a simple local time functional of a strong Markov process \(X^*\), called the genealogy-coding process. There are given a pathwise construction of \(X^*\) based on a Lévy process \(X\) with nonnegative jumps that does not drift to \(+\infty\) and whose Laplace exponent coincides with the branching mechanism, and an independent subordinator \(Y\) whose Laplace exponent coincides with the immigration mechanism. The local time process \(H\) is found to be a continuous state branching process with immigration. An analog of the classical Ray-Knight-Williams theorem for a general Lévy process with no negative jumps is derived.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 1885538
zbMATH Keywords: continuous state process with immigration
zbMATH Keywords: Lévy process
zbMATH Keywords: Brownian snake
reviewed by: V. V. Goryainov
MaRDI profile type: Publication
full work available at URL: s004400100155
OpenAlex ID: W2063347267
DOI: 10.1007/S004400100155
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1885538