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 thic kness of the distribution Prob( xi (0) is an element of (.)) close to its e ssential 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 o f variational problems. As a by-product, we establish Lifshitz tails for th e random Schrodinger operator -kappa Delta - xi at the bottom of its spectr um. In our class of xi distributions, the Lifshitz exponent ranges from d / 2 to infinity; the power law is typically accompanied by lower-order correc tions.