Smooth density field of catalytic super-Brownian motion
type: Item
instance of: scholarly article
title: Smooth density field of catalytic super-Brownian motion
zbMATH Open document ID: 0942.60082
author: Klaus Fleischmann
author: Achim Klenke
published in: The Annals of Applied Probability
publication date: 2000-08-21T00:00:00Z
review text: The authors consider a continuous super-Brownian motion (SBM) \((\rho_t)_{t\geq 0}\) in \(\mathbb{R}^d\) of dimensions \(d= 2,3\) with constant branching rate, and a catalytic SBM \((X^\rho_t)_{t\geq 0}\), that is, a continuous SBM in \(\mathbb{R}^d\) with ``local branching rate'' \(\rho_s(dx)\). The main result is that \(X^\rho_t\) is absolutely continuous with density function \(\xi^\rho_t\), and that there exists a version of the map \((t,z)\to \xi^\rho_t(z)\) which is \({\mathcal C}^\infty\) and solves the heat equation of the catalyst \(\rho\). This result and self-similarity are applied to show that in dimension \(d= 2\), if \(\rho\) and \(X^\rho\) are both started with a Lebesgue measure, then \(X^\rho_T\) converges in distribution as \(T\to\infty\) towards a random multiple of Lebesgue measure, the random multiple being \(\xi^\rho_1(0)\); thus persistent convergence holds.
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1346297
zbMATH Keywords: superprocess
zbMATH Keywords: absolutely continuous states
zbMATH Keywords: diffusion measures
zbMATH Keywords: persistence
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1029962743
OpenAlex ID: W1990681640
cites work: Collision Local Times and Measure-Valued Processes
cites work: Q3142396
cites work: Super-Brownian motions in higher dimensions with absolutely continuous measure states
cites work: A continuous super-Brownian motion in a super-Brownian medium
cites work: Longtime behavior of a branching process controlled by branching catalysts
cites work: Q3212092
cites work: The carrying dimension of a stochastic measure diffusion
cites work: Super-Brownian motion: Path properties and hitting probabilities
cites work: Super-mouvement brownien avec catalyse
cites work: Branching particle systems and superprocesses
cites work: Superprocesses and partial differential equations
cites work: Persistence of a two-dimensional super-Brownian motion in a catalytic medium
cites work: Absolute Continuity Results for Superprocesses with Some Applications
cites work: Measure-Valued Branching Diffusions with Singular Interactions
cites work: The Point Spectrum and Spectral Geometry for R<scp>iemann</scp>ian Manifolds with Cusps
cites work: A new approach to the single point catalytic super-Brownian motion
cites work: The longtime behavior of branching random walk in a catalytic medium
cites work: Stochastic partial differential equations for some measure-valued diffusions
cites work: Q4326617
cites work: The Hausdorff measure of the closed support of super-Brownian motion
cites work: Polar sets and multiple points for super-Brownian motion
DOI: 10.1214/AOAP/1029962743
Recommended article: The occupation density field for the catalytic super-Brownian motion
Recommended article: Q4840722
Recommended article: Super-mouvement brownien avec catalyse
Recommended article: Q4421370
Recommended article: Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst
Recommended article: Persistence of a two-dimensional super-Brownian motion in a catalytic medium
description: scientific article; zbMATH DE number 1346297