On the volume of the supercritical super-Brownian sausage conditioned on survival

Let α and β be positive constants. Let X be the supercritical super-Brownian motion corresponding to the evolution equation ut=12Δ+βu−αu2 in Rd and let Z be the binary branching Brownian-motion with branching rate β. For t⩾0, let R(t)=⋃s=0tsupp(X(s)), that is R(t) is the (accumulated) support of X up to time t. For t⩾0 and a>0, let Ra(t)=⋃x∈R(t)B̄(x,a). We call Ra(t) the super-Brownian sausage corresponding to the supercritical super-Brownian motion X. For t⩾0, let R̂(t)=⋃s=0tsupp(Z(s)), that is R̂(t) is the (accumulated) support of Z up to time t. For t⩾0 and a>0, let R̂a(t)=⋃x∈R(t)B̄(x,a). We call R̂a(t) the branching Brownian sausage corresponding to Z. In this paper we prove that limt→∞ 1tlog Eδ0[exp(−ν|Ra(t)|)| X survives] =limt→∞ 1tlog Êδ0 exp(−ν|R̂a(t)|)=−βfor all d⩾2 and all a,α,ν>0.