Gaussian asymptotics for a non-linear Langevin type equation driven by an $\alpha$-stable Lévy noise

Abstract

Consider the dynamics of a particle whose speed satisfies a one-dimensional stochastic differential equation driven by a small symmetric $\alpha$-stable Lévy process in a potential of the form a power function of exponent $\beta+1$. Two cases are studied: the noise could be path continuous, namely a standard Brownian motion, if $\alpha=2$, or pure jump Lévy process, if $\alpha\in(0,2)$. The main goal is to study a scaling limit of the position process with this speed, and one proves that the limit is Brownian in either case. This result is a generalization in some sense of the quadratic potential case studied recently by Hintze and Pavlyukevich.<br />