The Annals of Probability

Stochastic integral representation and regularity of the density for the exit measure of super-Brownian motion

Jean-François Le Gall and Leonid Mytnik

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Abstract

This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+β)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu=u1+β.

Article information

Source
Ann. Probab., Volume 33, Number 1 (2005), 194-222.

Dates
First available in Project Euclid: 11 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1108141725

Digital Object Identifier
doi:10.1214/009117904000000612

Mathematical Reviews number (MathSciNet)
MR2118864

Zentralblatt MATH identifier
1097.60033

Subjects
Primary: 60G57: Random measures
Secondary: 60G17: Sample path properties 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 35J65: Nonlinear boundary value problems for linear elliptic equations

Keywords
Super-Brownian motion exit measure stochastic integral representation martingale measure semilinear partial differential equation

Citation

Le Gall, Jean-François; Mytnik, Leonid. Stochastic integral representation and regularity of the density for the exit measure of super-Brownian motion. Ann. Probab. 33 (2005), no. 1, 194--222. doi:10.1214/009117904000000612. https://projecteuclid.org/euclid.aop/1108141725


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