Almost sure 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 kappa > 0 is a diffusion constant and xi is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t --> infinity.