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May 1999 Smooth density field of catalytic super-Brownian motion
Klaus Fleischmann, Achim Klenke
Ann. Appl. Probab. 9(2): 298-318 (May 1999). DOI: 10.1214/aoap/1029962743

Abstract

Given an (ordinary) super-Brownian motion (SBM) ϱ on Rd of dimension d=2,3, we consider a (catalytic) SBM Xϱ on Rd with "local branching rates" ϱs(dx). We show that Xtϱ is absolutely continuous with a density function ξtϱ, say. Moreover, there exists a version of the map (t,z)ξtϱ(z) which is C and solves the heat equation off the catalyst ϱ; more precisely, off the (zero set of) closed support of the time-space measure dsϱs(dx). Using self-similarity, we apply this result to give the following answer to an open problem on the long-term behavior of Xϱ in dimension d=2: If ϱ and Xϱ start with a Lebesgue measure, then does XTϱ converge (persistently) as T toward a random multiple of Lebesgue measure?

Citation

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Klaus Fleischmann. Achim Klenke. "Smooth density field of catalytic super-Brownian motion." Ann. Appl. Probab. 9 (2) 298 - 318, May 1999. https://doi.org/10.1214/aoap/1029962743

Information

Published: May 1999
First available in Project Euclid: 21 August 2002

zbMATH: 0942.60082
MathSciNet: MR1687355
Digital Object Identifier: 10.1214/aoap/1029962743

Subjects:
Primary: 60J80
Secondary: 60G57 , 60K35

Keywords: absolutely continuous states , diffusive measures , Persistence , smooth density field , Superprocess , time-space gaps of super-Brownian motion

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.9 • No. 2 • May 1999
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