Long-Time Tails in the Parabolic Anderson Model with Bounded Potential

We consider the parabolic Anderson problem ∂tu = κ∆u + ξu on (0,∞)× Zd with random i.i.d. potential ξ =( ξ(z))z∈Zd and the initial condition u(0,·) ≡ 1. Our main assumption is that esssupξ(0) = 0. In dependence of the thickness of the distribution Prob(ξ(0) ∈·) close to its essential supremum, we identify both the asymptotics of the moments of u(t,0) and the almost-sure asymptotics of u(t,0) as t →∞in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schro¨dinger operator −κ∆− ξ at the bottom of its spectrum. In our class of ξ-distributions, the Lifshitz exponent ranges from d 2 to ∞; the power law is typically accompanied by lower-order corrections.