Stable and Gaussian Fluctuation Limit of a Lévy-Driven Slow-Fast System with Polynomial Dissipation

Abstract

We study fluctuation limit of a slow-fast system driven by $\alpha$-stable Lévy noise with $1<\alpha <2.$ The slow component is generated by an odd polynomial function $f(y):=y^q,$ while in the fast component, the drift is $g(y):=-|y{|}^p\mathrm{sgn}(y)$ for some $p>0.$ Although the noise is given, the fluctuation limit is either a stable process or a Brownian motion, depending on both $p$ and $\alpha .$ The critical line between stable limit and Brownian limit is $q+1-p=\alpha /2.$