Convergence rate of Smoluchowski--Kramers approximation with stable Lévy noise
Qingming Zhao, Wei Wang
TL;DR
The paper analyzes the small-mass (large damping) limit of a Langevin equation driven by isotropic $\alpha$-stable Lévy noise via a slow-fast reformulation and a three-part splitting of the fast component for $\alpha\in(1,2)$. It derives an effective SK-type approximation and proves convergence results in either the uniform metric (with a rate $\varepsilon^\theta$ for $0\le\theta<1$) or the Skorokhod metric (for $\theta=0$), depending on noise regularity. The work highlights that, although convergence in Skorokhod topology holds and the limit solves $d\bar{U}(t)=f(\bar{U}(t))\,dt+dL(t)$, stronger uniform-in-$t$ $L^1$-type convergence may fail, and tightness arguments differ between the continuous-path and càdlàg settings. Overall, the paper extends SK-type averaging to Lévy-driven systems, providing convergence rates and a rigorous handling of stable noise via splitting techniques and tightness analysis.
Abstract
The small mass limit of the Langevin equation perturbed by $α$-stable Lévy noise is considered by rewriting it in the form of slow-fast system, and spliting the fast component into three parts, where $α\in(1,2)$. 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.