Tail probability of avoiding Poisson traps for branching Brownian motion

Abstract

We consider a branching Brownian motion Z with exponential branching times and general offspring distribution evolving in R-d, where Poisson traps are present. A Poisson trap configuration with radius a is defined to be the random subset K of R-d given by K = boolean OR(x)l(,is an element of supp)(M) (B) over bar (x(i), a), where M is a Poisson random measure on B(R-d) with constant trap intensity. Survival up to time t is defined to be the event {T > t) with T = inf{s >= 0 : Z(s)(K) > 0} being the first trapping time. Following the work of Englander (2000), Englander and den Hollander (2003), where strictly dyadic branching is considered, we consider here a general offspring distribution for Z and settle the problem of survival asymptotics for the system.