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 equationẍ ε t + |ẋ ε t | β =Ż 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 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.

Зародження І Розвиток Ідей Теорії Стохастичних Диференціальних Рівнянь В Українській Школі Математики

“…Низку надзвичайно цікавих тверджень такого типу одержано в роботах С. Я. Махна та його учнів, а також в роботах О. М. Кулика та А. Ю. Пилипенка разом з колегами (див. [1,7,18]).…”

Asymptotic analysis of dynamical systems driven by Poisson random measures with periodic sampling

“…In many other research areas, for instance, in physics, insurance risk models, and kinetic chemical reactions, one encounters many complex stochastic systems with jump noise, e.g., [KP19,Gil76,LY04]. The averaging principle for stochastic models driven by Poisson random measures has been studied extensively by many authors, see, for example, [Giv07, GP18, KP19, MYWM15, XDX11, XYW21], whereas for stochastic partial differential equations, we refer to [PZ07, DHI13, XML15, PXW17, Xu17].…”