Tail probability of avoiding Poisson traps for branching Brownian motion

We consider a branching Brownian motion Z with exponential branching times and general offspring distribution evolving in Rd, where Poisson traps are present. A Poisson trap configuration with radius a is defined to be the random subset K of Rd given by K=⋃xi∈supp(M)B̄(xi,a), where M is a Poisson random measure on B(Rd) with constant trap intensity. Survival up to time t is defined to be the event {T>t} with T=inf{s≥0:Zs(K)>0} being the first trapping time. Following the work of Engländer (2000), Engländer 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.