  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.
