Moment asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem .tu=..u+.(x)u on R+.Rd with initial condition u(0,x)=1. Here .(.) is a random shift-invariant potential having high .-like peaks on small islands. We express the second-order asymptotics of the pth moment (p.[1,.)) of u(t,0) as t.. in terms of a variational formula involving an asymptotic description of the rescaled shapes of these peaks via their cumulant generating function. This includes Gaussian potentials and high Poisson clouds.