Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment

Large time annealed path measure limits for a one-dimensional Brownian motion, with possibly a small drift, moving among "soft" Poissonian traps are considered. Limits with respect to both scaled and unscaled motions are derived. The results in both cases considered here agree with those shown before for the related model with "hard" traps. The proofs follow by generalizing previous techniques which identify a large clearing empty of traps in which typically the Brownian motion is confined. What is understood then, in the cases studied, is that under the annealed measure the soft traps organize to act in effect as their hard counterparts.