WIAS Preprint No. 491, (1999)

On the volume of the supercritical super-Brownian sausageconditioned on survival



Authors

  • Englaender, Janos

2010 Mathematics Subject Classification

  • 60J80 60J65 60D05

Keywords

  • super-Brownianmotion, super-Brownian sausage, branching Brownian motion, branchingBrownian sausage, Poissonian traps, hard obstacles

Abstract

Let $alpha$ and $beta$ be positive constants. Let $X$ be the supercritical super-Brownian motion corresponding to the evolution equation $u_t=frac12Delta+beta u-alpha u^2$ in $mathbb R^d$ and let $Z$ be the binary branching Brownian-motion with branching rate $beta$. For $tge 0$, let $R(t)=overlinebigcup_s=0^t mathrmsupp (X(s))$, that is $R(t)$ is the (accumulated) support of $X$ up to time $t$. For $tge 0$ and $a>0$, let $R^a(t)=bigcup_xin R(t)bar B(x,a).$ We call $R^a(t)$ the emphsuper-Brownian sausage corresponding to the supercritical super-Brownian motion $X$. For $tge 0$, let $hat R(t)=overlinebigcup_s=0^t mathrmsupp (Z(s))$, that is $hat R(t)$ is the (accumulated) support of $Z$ up to time $t$. For $tge 0$ and $a>0$, let $hat R^a(t)=bigcup_xin R(t)bar B(x,a).$ We call $hat R^a(t)$ the emphbranching Brownian sausage corresponding to $Z$. In this paper we prove that $$lim_ttoinftyfrac1tlog E_delta_0[exp(-nu R^a(t) ) , X mathrmsurvives]= lim_ttoinftyfrac1tloghat E_delta_0exp(-nu hat R^a(t) )= -beta,$$ for all $dge 2$ and all $a,alpha,nu>0$.

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