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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) $\varrho$ on $\mathbf{R}^d$ of dimension $d = 2, 3$, we consider a (catalytic) SBM $X^{\varrho}$ on $\mathbf{R}^d$ with "local branching rates" $\varrho_s(dx)$. We show that $X_t^{\varrho}$ is absolutely continuous with a density function $\xi_t^{\varrho}$, say. Moreover, there exists a version of the map $(t, z) \mapsto \xi_t^{\varrho}(z)$ which is $\mathscr{C}^{\infty}$ and solves the heat equation off the catalyst $\varrho$; more precisely, off the (zero set of) closed support of the time-space measure $ds\varrho_s(dx)$. Using self-similarity, we apply this result to give the following answer to an open problem on the long-term behavior of $X^{\varrho}$ in dimension $d = 2$: If $\varrho$ and $X^{\varrho}$ start with a Lebesgue measure, then does $X_T^{\varrho}$ converge (persistently) as $T \to \infty$ toward a random multiple of Lebesgue measure?

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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

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Vol.9 • No. 2 • May 1999
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