The Annals of Probability

Singularity of Super-Brownian Local Time at a Point Catalyst

Donald A. Dawson, Klaus Fleischmann, Yi Li, and Carl Mueller

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In a one-dimensional single point-catalytic continuous super-Brownian motion studied by Dawson and Fleischmann, the occupation density measure $\lambda^c$ at the catalyst's position $\mathcal{C}$ is shown to be a singular (diffuse) random measure. The source of this qualitative new effect is the irregularity of the varying medium $\delta_\mathcal{C}$ describing the point catalyst. The proof is based on a probabilistic characterization of the law of the Palm canonical clusters $\chi$ appearing in the Levy-Khintchine representation of $\lambda^\mathcal{C}$ in a historical process setting and the fact that these $\chi$ have infinite left upper density (with respect to Lebesgue measure) at the Palm time point.

Article information

Ann. Probab., Volume 23, Number 1 (1995), 37-55.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J65: Brownian motion [See also 58J65] 60G57: Random measures

Point catalytic medium critical branching super-Brownian local time occupation time occupation density measure-valued branching superprocess


Dawson, Donald A.; Fleischmann, Klaus; Li, Yi; Mueller, Carl. Singularity of Super-Brownian Local Time at a Point Catalyst. Ann. Probab. 23 (1995), no. 1, 37--55. doi:10.1214/aop/1176988375.

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