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

Abstract

For 0<α≤2, a super-α-stable motion X in $\mathsf{R}^{d}$ with branching of index 1+β∈(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.