Regularity and irregularity of (1 + β)-stable super-Brownian motion

This paper establishes the continuity of the density of (1 + β)-stable super-Brownian motion (0 < β < 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 ≥ 2 where the local time exists.