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

Let ξ=(ξt)ξ=(ξt) be a locally finite (2,β)(2,β)-superprocess in RdRd with β<1β<1 and d>2/βd>2/β. Then for any fixed t>0t>0, the random measure ξtξt can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the εε-neighborhoods of suppξt. This extends the Lebesgue approximation of Dawson–Watanabe superprocesses. Our proof is based on a truncation of (α,β)(α,β)-superprocesses and uses bounds and asymptotics of hitting probabilities.