On the exit measure of super-Brownian motion
type: Item
instance of: scholarly article
title: On the exit measure of super-Brownian motion
zbMATH Open document ID: 0801.60040
DOI: 10.1007/BF01199025
author: Romain Abraham
author: J.-F. Le Gall
publication date: 1994-11-21T00:00:00Z
review text: We study some properties of the exit measure of super-Brownian motion from a smooth domain \(D\) in \(\mathbb{R}^ d\). In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of \(\Delta u = u^ 2\) in \(D\), and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of \(\Delta u=u^ 2\) in \(D\) that tends to \(\infty\) on an open subset \(O\) of \(\partial D\) and to 0 on the complement in \(\partial D\) of the closure of \(O\). Our proofs use the path-valued process.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 639359
zbMATH Keywords: exit measure
zbMATH Keywords: super-Brownian motion
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: characterization of removable singularities at the boundary
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
cites work: Super-Brownian motion: Path properties and hitting probabilities
cites work: Historical processes
cites work: Branching particle systems and superprocesses
cites work: Path processes and historical superprocesses
cites work: A probabilistic approach to one class of nonlinear differential equations
cites work: Superdiffusion and parabolic nonlinear differential equations
cites work: Superprocesses and partial differential equations
cites work: Q1177210
cites work: Boundary singularities of solutions of some nonlinear elliptic equations
cites work: Brownian excursions, trees and measure-valued branching processes
cites work: A class of path-valued Markov processes and its applications to superprocesses
cites work: A Space-Time Property of a Class of Measure-Valued Branching Diffusions
cites work: Functions continuous and singular with respect to a Hausdorff measure
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 639359