Abstract.
We consider the parabolic Anderson problem ∂ t u = κΔu + ξ(x)u on ℝ+×ℝd with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞.
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Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000
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Gärtner, J., König, W. & Molchanov, S. Almost sure asymptotics for the continuous parabolic Anderson model. Probab Theory Relat Fields 118, 547–573 (2000). https://doi.org/10.1007/PL00008754
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DOI: https://doi.org/10.1007/PL00008754