Moment asymptotics for the continuous parabolic Anderson model

Abstract

We consider the parabolic Anderson problem $\partial_{t}u = \kappa\Delta u + \xi(x)u$ on $\mathbb{R}_+ \times \mathbb{R}^d$ with initial condition $u(0, x) = 1$. Here $\xi(\cdot)$ is a random shift-invariant potential having high $\delta$-like peaks on small islands. We express the second-order asymptotics of the $p$th moment $(p \in [1, \infty))$ of $u(t,0)$ as $t \to \infty$ 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.