Brownian survival and Lifshitz tail in perturbed lattice disorder

TL;DR

This paper analyzes the long-term behavior of Brownian motion survival probability in a lattice-distributed trap environment, revealing Lifshitz tail effects and intermittency estimates in related quantum and stochastic models.

Contribution

It provides the first detailed asymptotic analysis of survival probabilities in perturbed lattice traps, linking it to Lifshitz tails and intermittency in the Parabolic Anderson model.

Findings

Asymptotic formula for survival probability up to a constant

Identification of Lifshitz tail behavior in the density of states

Quantitative estimate for intermittency strength

Abstract

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.