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

Let and be positive constants. Let X be the supercritical super-Brownian motion corresponding to the evolution equation ut = 12 + u u2 in Rd and let Z be the binary branching Brownian-motion with branching rate . For t 0, let R(t) = Sts=0 supp(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) = Sx2R(t) B(x;a): We call Ra(t) the super-Brownian sausage corresponding to the supercritical super-Brownian motion X. For t 0, let R̂(t) = Sts=0 supp(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) = Sx2R(t) B(x;a): We call R̂a(t) the branching Brownian sausage corresponding to Z. In this paper we prove that lim t!1 1t logE 0 [exp( jRa(t)j)jX survives] = lim t!1 1t log Ê 0 exp( jR̂a(t)j) = ; for all d 2 and all a; ; > 0. 1