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

Abstract

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

Published in at http://dx.doi.org/10.1214/09-AOP501 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Stochastic integral representation and regularity of the density for the Exit measure of super-Brownian motion

Cited by in corpus (2)

• Multifractal analysis of superprocesses with stable branching in dimension one
• Equivalence of mean-field avalanches and branching diffusions: From the Brownian force model to the super-Brownian motion