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 λc at the catalyst's position Cis 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 Levy-Khintchine 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.