Non-Gaussian Limit Theorem for Non-Linear Langevin Equations Driven by Lévy Noise

Abstract

In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation ${\ddot{x}}_t^{\epsilon }+|{\dot{x}}_t^{\epsilon }{|}^{\beta }={\dot{Z}}_{\epsilon t}^{\epsilon }$, $\beta \in \mathbb{R}$, driven by symmetric non-Gaussian Lévy processes $Z^{\epsilon }$. 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 $t/\epsilon$ 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 Lévy 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 $\{Z^{\epsilon }{\}}_{\epsilon >0}$.

35 pages, 3 figures

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