Abstract
We consider the large-time behavior of the solution \( u:[0,\infty ) \times \mathbb{Z} \to [0,\infty ) \) to the parabolic Anderson problem ∂tu=κΔu+ξu with initial data u(0, ·)=1 and non-positive finite i.i.d. potentials \( (\xi (z))_{z \in \mathbb{Z}} \). Unlike in dimensions d≥2, the almost-sure decay rate of u(t, 0) as t→∞ is not determined solely by the upper tails of ξ(0); too heavy lower tails of ξ(0) accelerate the decay. The interpretation is that sites x with large negative ξ(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 [BK00] of this model in general dimension.
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Biskup, M., König, W. Screening Effect Due to Heavy Lower Tails in One-Dimensional Parabolic Anderson Model. Journal of Statistical Physics 102, 1253–1270 (2001). https://doi.org/10.1023/A:1004840328675
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DOI: https://doi.org/10.1023/A:1004840328675