Non-Gaussian Limit Theorem for Non-Linear Langevin Equations Driven by L\'evy Noise

Abstract

In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation $\ddot x^\varepsilon_t +|\dot x^\varepsilon_t|^\beta=\dot Z^\varepsilon_{\varepsilon t}$, $\beta\in\mathbb R$, driven by symmetric non-Gaussian L\'evy processes $Z^\varepsilon$. This equation describes the dynamics of a one-degree-of-freedom mechanical system subject to non-linear friction and noisy vibrations. For a compound Poisson noise, the process $x^\varepsilon$ on the macroscopic time scale $t/\varepsilon$ has a natural interpretation as a non-linear filter which responds to each single jump of the driving process. We prove that a system driven by a general symmetric L\'evy noise exhibits essentially the same asymptotic behaviour under the principal condition $\alpha+2\beta<4$, where $\alpha\in [0,2]$ is the ``uniform'' Blumenthal--Getoor index of the family $\{Z^\varepsilon\}_{\varepsilon>0}$.