 00382 Marek Biskup and Wolfgang Koenig
 Longtime tails in the parabolic Anderson model with bounded potential
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Oct 2, 00

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Abstract. 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 $u(0,\cdot)\equiv1$. Our main assumption is
that $\text{esssup}\xi(0)=0$. Depending on the thickness of the
distribution $\text{Prob}(\xi(0)\in\cdot)$ 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 Schr\"odinger 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
lowerorder corrections.
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