Extremes of the stochastic heat equation with additive Lévy noise

Abstract

We analyze the spatial asymptotic properties of the solution to the stochastic heat equation driven by an additive Lévy space-time white noise. For fixed time $t>0$ and space $x\in {\mathbb{R}}^d$ we determine the exact tail behavior of the solution both for light-tailed and for heavy-tailed Lévy jump measures. Based on these asymptotics we determine for any fixed time $t>0$ the almost-sure growth rate of the solution as $|x|\to \mathrm{\infty }$.