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

We consider the large-time behavior of the solution u: [0, proportional to ) x Z --> [0, proportional to) to the parabolic Anderson problem partial de rivative ,u = kappa Deltau + xiu with initial data u(0, proportional to) = 1 and non-positive finite i.i.d. potentials (xi (z))(z is an element ofZ). Unlike in dimensions d greater than or equal to 2, the almost-sure decay ra te u(t,0) as t --> proportional to is not determined solely by the upper ta ils of xi (0); too heavy lower tails of xi (0) accelerate the decay. The in terpretation is that sites x with large negative xi (x) hamper the mass flo w and hence screen off the influence of more favorable regions of the poten tial. The phenomenon is unique to d = 1. The result answers an open questio n from our previous study [BK00] of this model in general dimension.