Optimal local Hölder index for density states of superprocesses with $(1+β)$-branching mechanism

Abstract

For $0<α\leq2$, a super-$α$-stable motion $X$ in $\mathsf{R}^d$ with branching of index $1+β\in(1,2)$ is considered. Fix arbitrary $t>0$. If $d<α/β$, a dichotomy for the density function of the measure $X_t$ holds: the density function is locally Hölder continuous if $d=1$ and $α>1+β$ but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Hölder index.