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

In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation $\ddot x^\varepsilon_t +|\dot x^\varepsilon_t|^\beta=\dot Z^\varepsilon_{\varepsilon t}$,$\beta\in\mathbb R$,drivenbysymmetricnon-GaussianL\acute{e}vyprocesses$Z^\varepsilon$. 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 $x^{\epsilon }$ on the macroscopic time scale $t\mathrm{/}\epsilon$ 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 $\{Z^{\epsilon }{\}}_{\epsilon >0}$.