Long-time tails in the parabolic Anderson model with bounded potential

We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$on$(0,\infty)\times \Z^d$withrandomi\mathrm{.}i\mathrm{.}d\mathrm{.}potential$\xi=(\xi(z))_{z\in\Z^d}$andtheinitialcondition$u(0,\cdot)\equiv1$. 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 $t\to \mathrm{\infty }$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schr"odinger operator $-\kappa \mathrm{\Delta }-\xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d\mathrm{/}2$ to $\mathrm{\infty }$; the power law is typically accompanied by lower-order corrections.