Lebesgue approximation of -superprocesses

Let ξ = ( ξ t ) be a locally finite ( 2 , β ) -superprocess in R 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 ε -neighborhoods of 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.