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February 2000 Moment asymptotics for the continuous parabolic Anderson model
Jürgen Gärtner, Wolfgang König
Ann. Appl. Probab. 10(1): 192-217 (February 2000). DOI: 10.1214/aoap/1019737669

Abstract

We consider the parabolic Anderson problem $\partial_{t}u = \kappa\Delta u + \xi(x)u$ on $\mathbb{R}_+ \times \mathbb{R}^d$ with initial condition $u(0, x) = 1$. Here $\xi(\cdot)$ is a random shift-invariant potential having high $\delta$-like peaks on small islands. We express the second-order asymptotics of the $p$th moment $(p \in [1, \infty))$ of $u(t,0)$ as $t \to \infty$ in terms of a variational formula involving an asymptotic description of the rescaled shapes of these peaks via their cumulant generating function. This includes Gaussian potentials and high Poisson clouds.

Citation

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Jürgen Gärtner. Wolfgang König. "Moment asymptotics for the continuous parabolic Anderson model." Ann. Appl. Probab. 10 (1) 192 - 217, February 2000. https://doi.org/10.1214/aoap/1019737669

Information

Published: February 2000
First available in Project Euclid: 25 April 2002

zbMATH: 1171.60359
MathSciNet: MR1765208
Digital Object Identifier: 10.1214/aoap/1019737669

Subjects:
Primary: 60H25, 82C44
Secondary: 35B40, 60F10

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.10 • No. 1 • February 2000
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