Lebesgue approximation of $(2,\beta )$-superprocesses

Abstract

Let $\xi =({\xi }_t)$ be a locally finite $(2,\beta )$-superprocess in $\RR^d$ with $\beta <1$ and $d>2/\beta$. Then for any fixed $t>0$, the random measure ${\xi }_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\epsilon$-neighborhoods of $\mathrm{supp}{\xi }_t$. This extends the Lebesgue approximation of Dawson-Watanabe superprocesses. Our proof is based on a truncation of $(\alpha ,\beta )$-superprocesses and uses bounds and asymptotics of hitting probabilities.

arXiv admin note: text overlap with arXiv:0901.2840