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April 2001 Long-Time Tails in The Parabolic Anderson Model with Bounded Potential
Marek Biskup, Wolfgang König
Ann. Probab. 29(2): 636-682 (April 2001). DOI: 10.1214/aop/1008956688


We consider the parabolic Anderson problem $\partial_t u = \kappa\Delta u + \xi u$ on $(0,\infty) \times \mathbb{Z}^d$ with random i.i.d. potential $\xi = (\xi(z))_{z\in \mathbb{Z}^d}$ and the initial condition $u(0,\cdot) \equiv 1$. Our main assumption is that $\mathrm{esssup} \xi(0)=0$. Depending on the thickness of the distribution $\mathrm{Prob} (\xi(0) \in \cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t, 0)$ and the almost­sure asymptotics of $u(t, 0)$ as $t \to \infty$ in terms of variational problems. As a by­product, we establish Lifshitz tails for the random Schrödinger operator $-\kappa \Delta - \xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.


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Marek Biskup. Wolfgang König. "Long-Time Tails in The Parabolic Anderson Model with Bounded Potential." Ann. Probab. 29 (2) 636 - 682, April 2001.


Published: April 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1018.60093
MathSciNet: MR1849173
Digital Object Identifier: 10.1214/aop/1008956688

Primary: 60F10 , 82B44
Secondary: 35B40 , 35K15

Keywords: almost­sure asymptotics , Dirichlet eigenvalues , Intermittency , large deviations , Lifshitz tails , Moment asymptotics , Parabolic Anderson model , percolation

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2001
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