On the filtration of historical Brownian motion
type: Item
instance of: scholarly article
title: On the filtration of historical Brownian motion
zbMATH Open document ID: 0816.60044
author: Martin T. Barlow
author: Edwin Perkins
published in: The Annals of Probability
publication date: 1995-07-06T00:00:00Z
review text: The historical Brownian motion \(H\) is a process which takes its values in \(M_ F(C)\), the set of finite measures on \(C\), where \(C= C([0,\infty[, R^ d)\). If \(\nu\in M_ F(C)\), let \(\nu_ t\) be the image of \(\nu\) by the map \(y\to y(t)\). The super-Brownian motion \(X\) is defined by: \(X_ t\) is the image of \(H_ t\) by \(\nu\to \nu_ t\). The main result is the following: if \(d\geq 5\), then \({\mathcal F}^ X_ t= {\mathcal F}^ H_ t\) \(\forall t\geq 0\); moreover if \(d= 1\), then \({\mathcal F}^ X_ t\neq {\mathcal F}^ H_ t\). The proof is based on the fact that if \(d\geq 4\), a super- Brownian motion in \(M_ F(R^ d)\) started at \(m\neq 0\) does not intersect an independent \(R^ d\)-valued Brownian motion. If \(d= 2, 3\), the authors claim that \({\mathcal F}^ X_ t\neq {\mathcal F}^ H_ t\). The question is open when \(d= 4\).
reviewed by: Pierre Vallois
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G07
zbMATH DE Number: 731983
zbMATH Keywords: historical Brownian motion
Wikidata QID: Q101580012
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988603
OpenAlex ID: W1998788921
DOI: 10.1214/AOP/1176988603
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description: scientific article; zbMATH DE number 731983