WIAS Preprint No. 56, (1993)

Singularity of super-Brownian local time at a point catalyst.


  • Dawson, Donald A.
  • Fleischmann, Klaus
  • Li, Yi
  • Müller, Carl

2010 Mathematics Subject Classification

  • 60J80 60J65 60G57


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




In a one-dimensional single point-catalytic continuous super-Brownian motion studied by Dawson and Fleischmann (1993), the occupation density measure λc at the catalyst's position c is shown to be a singular (diffuse) random measure. The source of this qualitative new effect is the irregularity of the varying medium δc describing the point catalyst. The proof is based on a probabilistic characterization of the law of the Palm canonical clusters χ appearing in the Lévy-Khinchin representation of λc in a historical process setting and the fact that these χ have infinite left upper density (with respect to Lebesgue measure) at the Palm time point.

Appeared in

  • The Annals of Probability 23(1)(1995), pp. 37-55

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