Long-time tails in the parabolic Anderson model with bounded potential

We consider the parabolic Anderson problem ∂ t u = κΔu + ξu on (0, ∞) × Z d with random i.i.d. potential ξ = (ξ(z)) z ∈ Zd and the initial condition u(0,.) ≡ 1. Our main assumption is that esssup ξ(0) = 0. Depending on 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 Schrodinger operator - κΔ - ξ at the bottom of its spectrum. In our class of distributions, the Lifshitz exponent ranges from d/2 to ∞; the power law is typically accompanied by lower-order corrections.