Singularity of super-Brownian local time at a point catalyst
In a one-dimensional single point-catalytic continuous super-Brownian motion studied by \textit{D. A. Dawson} and \textit{K. Fleischmann} [Stochastic Processes Appl. 49, No. 1, 3-40 (1994; Zbl 0796.60087)] and \textit{K. Fleischmann} [in: Measure-valued processes, stochastic partial differential equations, and interacting systems. CRM Proc. Lect. Notes 5, 99-110 (1994; Zbl 0802.60076)], the occupation density measure \(\lambda^{\mathcal C}\) at the catalyst's position \({\mathcal C}\) is shown to be a singular (diffuse) random measure. The source of this qualitatively 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 Lévy-Khinchin 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.
scientific article; zbMATH DE number 796531