00-288 Marek Biskup and Wolfgang Koenig

Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model (1368K, LaTeX & Postscript) Jul 10, 00

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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.

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