The Annals of Applied Probability

Smooth density field of catalytic super-Brownian motion

Klaus Fleischmann and Achim Klenke

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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?

Article information

Source
Ann. Appl. Probab., Volume 9, Number 2 (1999), 298-318.

Dates
First available in Project Euclid: 21 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1029962743

Digital Object Identifier
doi:10.1214/aoap/1029962743

Mathematical Reviews number (MathSciNet)
MR1687355

Zentralblatt MATH identifier
0942.60082

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G57: Random measures 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

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

Citation

Fleischmann, Klaus; Klenke, Achim. Smooth density field of catalytic super-Brownian motion. Ann. Appl. Probab. 9 (1999), no. 2, 298--318. doi:10.1214/aoap/1029962743. https://projecteuclid.org/euclid.aoap/1029962743


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