Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential

Abstract

We consider the parabolic Anderson model (PAM) ${\partial }_{t}u=\frac{1}{2}\hat{I}”u+\hat{I}¾u$ in ${\mathbb{R}}^2$ with a Gaussian (space) white-noise potential $\hat{I}¾$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $\log U(t)\sim \ddot{I}‡t\log t$ for $t\to \infty$, with the deterministic constant $\ddot{I}‡$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour $\hat{\mathbf{I}}{»}_1(Q_t)\sim \ddot{I}‡\log t$ of the principal eigenvalue $\hat{\mathbf{I}}{»}_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t=[-\frac{t}{2},\frac{t}{2}{]}^2$.