SINGULARITY OF SUPER-BROWNIAN LOCAL, TIME AT A POINT CATALYST

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 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(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(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.