From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion in the Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion in traps distributed according to a random perturbation of the lattice.