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 $\esssup \xi(0)=0$.
Depending on the thickness of the distribution $\Prob (\xi(0) \in \cdot)$ 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.
Publié le : 2001-04-14
Classification:
Parabolic Anderson model,
intermittency,
Lifshitz tails,
moment asymptotics,
almostsure asymptotics,
large deviations,
Dirichlet eigenvalues,
percolation,
60F10,
82B44,
35B40,
35K15
@article{1008956688,
author = {Biskup, Marek and and K\"onig, Wolfgang},
title = {LongTime Tails in The Parabolic Anderson Model with Bounded
Potential},
journal = {Ann. Probab.},
volume = {29},
number = {1},
year = {2001},
pages = { 636-682},
language = {en},
url = {http://dml.mathdoc.fr/item/1008956688}
}
Biskup, Marek and; König, Wolfgang. LongTime Tails in The Parabolic Anderson Model with Bounded
Potential. Ann. Probab., Tome 29 (2001) no. 1, pp. 636-682. http://gdmltest.u-ga.fr/item/1008956688/