Moment asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem partial derivative (t)u = kappa Deltau + xi (x)u on R+ x R-d with initial condition u(0, x) = 1. Here xi(.) is a random shift-invariant potential having high delta -like peaks on sma ll islands. We express the second-order asymptotics of the pth moment (p is an element of [1, infinity)) of u(t, 0) as t --> infinity in terms of a va riational formula involving an asymptotic description of the rescaled shape s of these peaks via their cumulant generating function. This includes Gaus sian potentials and high Poisson clouds.