Extremes of the stochastic heat equation with additive L\'evy noise

TL;DR

This paper investigates the spatial behavior of solutions to the stochastic heat equation driven by additive Lévy noise, providing precise tail asymptotics and growth rates for different Lévy jump measures.

Contribution

It offers the first detailed analysis of the tail behavior and almost-sure growth rates of solutions with general Lévy noise in the stochastic heat equation.

Findings

Exact tail asymptotics for light-tailed Lévy measures

Exact tail asymptotics for heavy-tailed Lévy measures

Determination of almost-sure growth rates as |x| approaches infinity

Abstract

We analyze the spatial asymptotic properties of the solution to the stochastic heat equation driven by an additive L\'evy 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 L\'evy jump measures. Based on these asymptotics we determine for any fixed time $t> 0$ the almost-sure growth rate of the solution as $|x| \to \infty$.