The Annals of Applied Probability

Moment asymptotics for the continuous parabolic Anderson model

Jürgen Gärtner and Wolfgang König

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 10, Number 1 (2000), 192-217.

Dates
First available in Project Euclid: 25 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019737669

Digital Object Identifier
doi:10.1214/aoap/1019737669

Mathematical Reviews number (MathSciNet)
MR1765208

Zentralblatt MATH identifier
1171.60359

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60F10: Large deviations 35B40: Asymptotic behavior of solutions

Keywords
Parabolic Anderson problem random medium large deviations moment asymptotics heat equation with random potential

Citation

Gärtner, Jürgen; König, Wolfgang. Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10 (2000), no. 1, 192--217. doi:10.1214/aoap/1019737669. https://projecteuclid.org/euclid.aoap/1019737669


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