The Cutoff Phenomenon in Wasserstein Distance for Nonlinear Stable Langevin Systems with Small Lévy Noise

• Published in: Journal of dynamics and differential equations Vol. 36; no. 1; pp. 251 - 278
• Main Authors: Barrera, G., Högele, M. A., Pardo, J. C.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Dissipation; Gradient flow; Linearity; Mathematics; Mathematics and Statistics; Nonlinear differential equations; Nonlinear systems; Ordinary Differential Equations; Partial Differential Equations; 60H10; Nonlinear Langevin dynamics; 60G51; Nonstandard properties of the Wasserstein distance; 37A25; Exponential ergodicity; 15A16; Lévy processes; Cutoff phenomenon
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-022-10138-1

This article establishes the cutoff phenomenon in the Wasserstein distance for systems of nonlinear ordinary differential equations with a dissipative stable fixed point subject to small additive Markovian noise. This result generalizes the results shown in Barrera, Högele, Pardo (EJP2021) in a more restrictive setting of Blumenthal-Getoor index α > 3 / 2 to the formulation in Wasserstein distance, which allows to cover the case of general Lévy 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, Högele, 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 nonlinear Fermi–Pasta–Ulam–Tsingou gradient flow and dissipative nonlinear oscillators subject to small (and possibly degenerate) Brownian or arbitrary α -stable noise.