Limit theorems for the one-dimensional parabolic Anderson model with white noise potential

Abstract

We consider the parabolic Anderson model $\partial_t u=\partial_x^2 u+\xi u$ on $\mathbb{R}_+\times\mathbb{R}$ with $u(0,\cdot)\equiv 1$, where $\xi$ is a spatial white noise. We study the long-time behavior of the spatial integral $U(t):=\int_{-L(t)/2}^{L(t)/2}u(t,x)\,dx$, where $L(t)=\exp(\alpha^3 t^3/24)$ with $\alpha>0$. We establish a weak law of large numbers for $\alpha>1$ and a central limit theorem for $\alpha>2$. Moreover, for every $\alpha\in(0,2)$, we show that $U(t)$ converges in distribution, after explicit centering and scaling, to a totally asymmetric $\alpha$-stable law. To the best of our knowledge, these are the first stable limit laws for the continuous parabolic Anderson model. In addition, we establish two spectral results of independent interest for the one-dimensional Anderson Hamiltonian on a growing interval. They are the main ingredients of the proofs. The first gives the lower-tail asymptotics of the lowest eigenvalue, including the exact prefactor. The second shows that, on this lower-tail event, the $L^1$ norm of the corresponding eigenfunction concentrates around an explicit deterministic value.