The cutoff phenomenon for the stochastic heat and the wave equation subject to small L\'evy noise

TL;DR

This paper extends the cutoff phenomenon to the stochastic heat and wave equations with small L9vy noise, revealing profile and window cutoff behaviors in infinite dimensions, with new results for multiplicative noise cases.

Contribution

It generalizes the cutoff phenomenon to infinite-dimensional stochastic PDEs with L9vy noise, including new insights for multiplicative noise in the heat equation.

Findings

Profile cutoff for stochastic heat equation with additive noise

Window cutoff for stochastic wave equation with additive noise

Profile cutoff for stochastic heat equation with multiplicative noise

Abstract

This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and L\'evy noises in the Wasserstein distance. For the additive noise case, we obtain analogous infinite dimensional results to the respective finite dimensional cases obtained recently by Barrera, H\"ogele and Pardo (JSP2021), that is, the (stronger) profile cutoff phenomenon for the stochastic heat equation and the (weaker) window cutoff phenomenon for the stochastic wave equation. For the multiplicative noise case, which is studied in this context for the first time, the stochastic heat equation also exhibits profile cutoff phenomenon, while for the stochastic wave equation the methods break down due to the lack of symmetry. The methods rely strongly on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the stochastic solution flows in terms of stochastic exponentials.