Convergence rate of Smoluchowski--Kramers approximation with stable L\'{e}vy noise

TL;DR

This paper investigates the convergence rate of the Smoluchowski--Kramers approximation for Langevin equations driven by alpha-stable Lévy noise, providing explicit rates depending on noise regularity.

Contribution

It derives the convergence rate of the approximation in the small mass limit for Langevin equations with stable Lévy noise, considering different noise regularities.

Findings

Convergence rate established in the uniform metric.

Convergence rate established in the Skorokhod metric.

Approximation derived for the small mass limit with stable Lévy noise.

Abstract

The small mass limit of the Langevin equation perturbed by $\alpha$-stable L\'{e}vy noise is considered by rewriting it in the form of slow-fast system, and spliting the fast component into three parts, where $\alpha\in(1,2)$. By exploring the three parts respectively, the approximation equation is derived. The convergence is either in the sense of uniform metric or in the sense of Skorokhod metric, depending on how regular the noise is. In the former case, we obtain the convergence rate.