## The Annals of Applied Probability

### Moment asymptotics for the continuous parabolic Anderson model

#### 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

https://projecteuclid.org/euclid.aoap/1019737669

Digital Object Identifier
doi:10.1214/aoap/1019737669

Mathematical Reviews number (MathSciNet)
MR1765208

Zentralblatt MATH identifier
1171.60359

#### 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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