Non-Gaussian limit theorem for non-linear Langevin equations driven by Lévy noise
A Kulik, I Pavlyukevich - 2019 - projecteuclid.org
2019•projecteuclid.org
In this paper, we study the small noise behaviour of solutions of a non-linear second order
Langevin equation ̈x^ε_t+|̇x^ε_t|^β=̇Z^ε_εt, β∈R, driven by symmetric non-Gaussian
Lévy processes Z^ε. 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^ε on the macroscopic time scale t/ε has a natural interpretation
as a non-linear filter which responds to each single jump of the driving process. We prove …
Langevin equation ̈x^ε_t+|̇x^ε_t|^β=̇Z^ε_εt, β∈R, driven by symmetric non-Gaussian
Lévy processes Z^ε. 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^ε on the macroscopic time scale t/ε has a natural interpretation
as a non-linear filter which responds to each single jump of the driving process. We prove …
Abstract
In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation , , driven by symmetric non-Gaussian Lévy processes . 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 on the macroscopic time scale 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 , where is the “uniform” Blumenthal–Getoor index of the family .
Project Euclid