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.
Publié le : 2000-02-14
Classification:
Parabolic Anderson problem,
random medium,
large deviations,
moment asymptotics,
heat equation with random potential,
60H25,
82C44,
60F10,
35B40
@article{1019737669,
author = {G\"artner, J\"urgen and K\"onig, Wolfgang},
title = {Moment asymptotics for the continuous parabolic Anderson
model},
journal = {Ann. Appl. Probab.},
volume = {10},
number = {2},
year = {2000},
pages = { 192-217},
language = {en},
url = {http://dml.mathdoc.fr/item/1019737669}
}
Gärtner, Jürgen; König, Wolfgang. Moment asymptotics for the continuous parabolic Anderson
model. Ann. Appl. Probab., Tome 10 (2000) no. 2, pp. 192-217. http://gdmltest.u-ga.fr/item/1019737669/