Lebesgue approximation of $(2,β)$-superprocesses

Abstract

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