Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model

TL;DR

This paper investigates how heavy lower tails in the potential affect the decay of solutions in the one-dimensional parabolic Anderson model, revealing a unique screening effect not present in higher dimensions.

Contribution

It demonstrates that heavy negative tails in the potential accelerate decay in 1D, addressing an open question about the influence of lower tail behavior on the model.

Findings

Heavy lower tails cause faster decay of the solution in 1D.

Sites with large negative potential hinder mass flow and cause screening.

The phenomenon is specific to one-dimensional cases.

Abstract

We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(\xi(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $\xi(0)$; too heavy lower tails of $\xi(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $\xi(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study \cite{BK00} of this model in general dimension.