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

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

2008-05-26

arxiv.org/abs/0805.3914v4

Annals of Probability 2010, Vol. 38, No. 3, 1180-1220

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP501

For $0<\alpha\leq2$, 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 H\"{o}lder continuous if $d=1$ and $\alpha>1+\beta$ but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local H\"{o}lder index.

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