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

We consider the parabolic Anderson problem partial derivative (t)u = kappa Deltau + xiu on (0, infinity) x Z(d) with random i.i.d. potential xi = (xi (z))(z is an element ofZ)(d) and the initial condition u(0, (.)) equivalent to 1. Our main assumption is that esssup xi (0) = 0. Depending on the thickness of the distribution Prob( xi (0) is an element of (.)) 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 --> infinity, in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrodinger operator -kappa Delta - xi at the bottom of its spectrum. In our class of xi distributions, the Lifshitz exponent ranges from d /2 to infinity; the power law is typically accompanied by lower-order corrections.