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

2017/07/06 by Alexei Kulik, Kulik, Alexei, Ilya Pavlyukevich +1
Mathematics · #60H10 #60J25 #70F40 #70L05 #FOS: Mathematics #Primary 60F05 #Probability (math.PR) #Secondary 60G51 #math.PR #msc:60F05 #msc:60G51 #msc:60H10 #msc:60J25 #msc:70F40 #msc:70L05

paper · pdf · doi:10.48550/arxiv.1707.01958

35 pages, 3 figures

arxiv created 2018/07/20 · arxiv updated 2018/07/23

Abstract

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_{ε t}, β∈\mathbb 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 that a system driven by a general symmetric Lévy noise exhibits essentially the same asymptotic behaviour under the principal condition α+2β<4, where α∈ [0,2] is the ``uniform'' Blumenthal--Getoor index of the family \Z^ε\_ε>0.

Related

• Suppression of oscillations by Levy noise
• Parameter Estimation for the Langevin Equation with Stationary-Increment Gaussian Noise
• The second fluctuation-dissipation theorem for the generalized Langevin equation
• Asymptotic stability and cut-off phenomenon for the underdamped Langevin dynamics
• Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and Lévy noise