Regularity and irregularity of $\bolds{(1+\beta)}$-stable super-Brownian motion

Abstract

This paper establishes the continuity of the density of $(1+\beta)$-stable super-Brownian motion $(0<\beta<1)$ for fixed times in $d=1$, and local unboundedness of the density in all higher dimensions where it exists. We also prove local unboundedness of the density in time for a fixed spatial parameter in any dimension where the density exists, and local unboundedness of the occupation density (the local time) in the spatial parameter for dimensions $d\geq2$ where the local time exists.