Singularity of super-Brownian local time at a point catalyst
type: Item
instance of: scholarly article
title: Singularity of super-Brownian local time at a point catalyst
zbMATH Open document ID: 0830.60079
author: Donald A. Dawson
author: Klaus Fleischmann
author: Yi Li
author: Carl Mueller
published in: The Annals of Probability
publication date: 1996-01-28T00:00:00Z
review text: In a one-dimensional single point-catalytic continuous super-Brownian motion studied by \textit{D. A. Dawson} and \textit{K. Fleischmann} [Stochastic Processes Appl. 49, No. 1, 3-40 (1994; Zbl 0796.60087)] and \textit{K. Fleischmann} [in: Measure-valued processes, stochastic partial differential equations, and interacting systems. CRM Proc. Lect. Notes 5, 99-110 (1994; Zbl 0802.60076)], the occupation density measure \(\lambda^{\mathcal C}\) at the catalyst's position \({\mathcal C}\) is shown to be a singular (diffuse) random measure. The source of this qualitatively new effect is the irregularity of the varying medium \(\delta_{\mathcal C}\) describing the point catalyst. The proof is based on a probabilistic characterization of the law of the Palm canonical clusters \(\chi\) appearing in the Lévy-Khinchin representation of \(\lambda^{\mathcal C}\) in a historical process setting and the fact that these \(\chi\) have infinite left upper density (with respect to Lebesgue measure) at the Palm time point.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 796531
zbMATH Keywords: point catalytic medium
zbMATH Keywords: critical branching
zbMATH Keywords: super-Brownian local time
zbMATH Keywords: occupation time
zbMATH Keywords: occupation density
zbMATH Keywords: measure-valued branching
zbMATH Keywords: superprocess
zbMATH Keywords: nonlinear partial differential equation
reviewed by: Valentin Topchii
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988375
OpenAlex ID: W2138562930
DOI: 10.1214/AOP/1176988375
description: scientific article; zbMATH DE number 796531