Brownian survival and Lifshitz tail in perturbed lattice disorder

We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of the traps is given by independent displacements of the lattice points. We determine the long time asymptotics of the logarithm of the survival probability up to a multiplicative constant. As applications, we show the Lifshitz tail effect of the density of states of the associated random Schr\"{o}dinger operator and derive a quantitative estimate for the strength of intermittency in the Parabolic Anderson problem.