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

Abstract

We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 \Delta u + \xi u$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $\log U(t)\sim \chi t \log t$ for $t \to \infty$, with the deterministic constant $\chi$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour $\boldsymbol \lambda_1(Q_t)\sim\chi\log t$ of the principal eigenvalue $\boldsymbol\lambda_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t= [-\frac{t}{2},\frac{t}{2}]^2$.