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

Let $\alpha$ and $\beta$ be positive constants. Let $X$ be the supercritical super-Brownian motion corresponding to the evolution equation $u_t=\frac{1}{2}\Delta+\beta u-\alpha u^2$ in $\mathbb R^d$ and let $Z$ be the binary branching Brownian-motion with branching rate $\beta$.\ For $t\ge 0$, let $R(t)=\overline{\bigcup_{s=0}^t \mathrm{supp} (X(s))}$, that is $R(t)$ is the (accumulated) support of $X$ up to time $t$. For $t\ge 0$ and $a>0$, let $R^a(t)=\bigcup_{x\in R(t)}\bar B(x,a).$ We call $R^a(t)$ the \emph{super-Brownian sausage} corresponding to the supercritical super-Brownian motion $X$. For $t\ge 0$, let $\hat R(t)=\overline{\bigcup_{s=0}^t \mathrm{supp} (Z(s))}$, that is $\hat R(t)$ is the (accumulated) support of $Z$ up to time $t$. For $t\ge 0$ and $a>0$, let $\hat R^a(t)=\bigcup_{x\in R(t)}\bar B(x,a).$ We call $\hat R^a(t)$ the \emph{branching Brownian sausage} corresponding to $Z$. In this paper we prove that $$\lim_{t\to\infty}\frac{1}{t}\log E_{\delta_0}[\exp(-\nu R^a(t) ) \, X\ \mathrm{survives}]= \lim_{t\to\infty}\frac{1}{t}\log\hat E_{\delta_0}\exp(-\nu \hat R^a(t) )= -\beta,$$ for all $d\ge 2$ and all $a,\alpha,\nu>0$