Properties of states of super-[alpha]-stable motion with branching of index 1 + [beta]

Abstract

It has been well-known for a long time that the measure states of the process in the title are absolutely continuous at any fixed time provided that the dimension of space is small enough. However, besides the very special case of one-dimensional continuous super-Brownian motion, properties of the related density functions were not well understood. Only in 2003, Mytnik and Perkins citeMytnikPerkins2003 revealed that in the Brownian motion case and if the branching is discontinuous, there is a dichotomy for the densities: Either there are continuous versions of them, or they are locally unbounded. We recently showed, that the same type of fixed time dichotomy holds also in the case of discontinuous motion. Moreover, the continuous versions are locally Hölder continuous, and we determined the optimal index for them. Finally, we determine the optimal index of Hölder continuity at given space points which is strictly larger than the optimal index of local Hölder continuity.