Marek Biskup and Wolfgang Koenig
Long-time tails in the parabolic Anderson model with bounded potential
(152K, LaTeX2e+times macro)
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 almost-sure asymptotics of $u(t,0)$ as
$t\to\infty$ in terms of variational problems. As a by-product,
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
lower-order corrections.