Superprocesses and projective limits of branching Markov process
type: Item
instance of: scholarly article
title: Superprocesses and projective limits of branching Markov process
zbMATH Open document ID: 0724.60090
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1991-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77401
review text: The author presents an a.s. limit theorem to construct a class of measure-valued processes defined by Watanabe and Dawson [cf. \textit{S. Watanabe}, J. Math. Kyoto Univ. 8, 141-167 (1968; Zbl 0159.462)]. Consider a family of branching Markov processes indexed by \(\tau \in {\mathbb{R}}^+\). The paths start at x with a number of points following a Poisson law of mean \(\psi \tau^{-1/\beta}\), and points move independently according to semigroup \(P_ t\) and branch at exponential times of mean \(\tau\) according to a reproduction generating function \(\sum q_ ku^ k=(1-u)^{1+\beta}/(1+\beta)+u\) (\(\beta\in]0,1])\). Denoting by \(Z^{\tau}_ t\) the induced multiple-valued process and taking a projective limit, the author proves that, using a martingale approach, the measure \(\psi^{-1}\epsilon^{1/\beta}\tilde Z^{\epsilon}_{t-\epsilon}P_{\epsilon}\) converges a.s. to a measure \(\mu_ t\) as \(\epsilon\downarrow 0\) and that \(\mu_ t\) is a Markov process. If \(U_ tf(x)=-Log(E(\exp (-\mu_ t(f)))\) for any positive bounded measurable function f, then \(U_ t(f)\) is uniquely determined by the equation \[ U_ t(f)=P_ tf-\psi^{-\beta}\int^{t}_{0}P_ s(U_{t-s}(f)^{1+\beta})ds/(1+\beta). \]
reviewed by: Chengxun Wu
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 4194892
zbMATH Keywords: Galton-Watson process
zbMATH Keywords: superprocess
zbMATH Keywords: branching Markov processes
zbMATH Keywords: semigroup
author: Yves Le Jan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4194892