On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

TL;DR

This paper investigates the growth and spatial positioning of high peaks in solutions to non-linear stochastic heat and wave equations driven by Lévy processes, revealing linear growth in peak distance over time under certain conditions.

Contribution

It establishes the linear growth rate of the farthest high peaks of solutions to stochastic heat and wave equations driven by Lévy processes, a novel insight in this area.

Findings

Peak distances grow linearly with time

Solutions' moments grow exponentially over time

Results extend from heat to wave equations

Abstract

We study a family of non-linear stochastic heat equations in (1+1) dimensions, driven by the generator of a L\'evy process and space-time white noise. We assume that the underlying L\'evy process has finite exponential moments in a neighborhood of the origin and that the initial condition has exponential decay at infinity. Then we prove that under natural conditions on the non-linearity: (i) The absolute moments of the solution to our stochastic heat equation grow exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the non-linear stochastic heat equation under the present setting. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.