The cutoff phenomenon in Wasserstein distance for nonlinear stable Langevin systems with small L\'evy noise

TL;DR

This paper proves the cutoff phenomenon in Wasserstein distance for nonlinear Langevin systems with small Lévy noise, extending previous results to more general Lévy processes and providing precise asymptotics.

Contribution

It generalizes the cutoff phenomenon results to systems with general Lévy noise in Wasserstein distance, using novel control of nonlinear flow errors and Wasserstein linearity properties.

Findings

Establishes cutoff phenomenon in Wasserstein distance for nonlinear systems with Lévy noise.

Extends previous results to general Lévy processes with moments.

Provides precise asymptotics of nonlinear flow and Wasserstein distance behavior.

Abstract

This article establishes the cutoff phenomenon in the Wasserstein distance for systems of nonlinear ordinary differential equations with a unique coercive stable fixed point subject to general additive Markovian noise in the limit of small noise intensity. This result generalizes the results shown in Barrera, H\"ogele, Pardo (EJP2021) in a more restrictive setting of Blumenthal-Getoor index $\alpha>3/2$ to the formulation in Wasserstein distance, which allows to cover the case of general L\'evy processes with some given moment. The main proof techniques are based on the close control of the errors in a version of the Hartman-Grobman theorem and the adaptation of the linear theory established in Barrera, H\"ogele, Pardo (JSP2021). In particular, they rely on the precise asymptotics of the nonlinear flow and the nonstandard shift linearity property of the Wasserstein distance, which is established by the authors in (JSP2021). Main examples are the Fermi-Pasta-Ulam-Tsingou gradient flow and coercive nonlinear oscillators subject to small (and possibly degenerate) Brownian or arbitrary $\alpha$-stable noise.