Singularity of Super-Brownian Local Time at a Point Catalyst

Abstract

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.