Regularity and irregularity of superprocesses with $(1+\beta)$-stable branching mechanism

Abstract

We would like to give an overview of results on regularity, or better to say "irregularity", properties of densities at fixed times of super-Brownian motion with $(1+\beta)$-stable branching for $\beta<1$. First, the following dichotomy for the density is shown: it is continuous in the dimension $d=1$ and locally unbounded in all higher dimensions where it exists. Then in $d=1$ we determine pointwise and local H\"older exponents of the density, and calculate the multifractal spectrum corresponding to pointwise H\"older exponents.