Eon, Richard; Gradinaru, Mihai Gaussian asymptotics for a non-linear Langevin type equation driven by an \(\alpha\)-stable Lévy noise. (English) Zbl 1328.60135 Electron. J. Probab. 20, Paper No. 100, 19 p. (2015). Summary: Consider the dynamics of a particle whose speed satisfies a one-dimensional stochastic differential equation driven by a small symmetric \(\alpha\)-stable Lévy process in a potential of the form of a power function of exponent \(\beta+1\). Two cases are studied: the noise could be path continuous, namely a standard Brownian motion, if \(\alpha=2\), or pure jump Lévy process, if \(\alpha\in(0,2)\). The main goal is to study a scaling limit of the position process with this speed, and one proves that the limit is Brownian in either case. This result is a generalization in some sense of the quadratic potential case studied recently by R. Hintze and I. Pavlyukevich [Bernoulli 20, No. 1, 265–281 (2014; Zbl 1309.60059)]. Cited in 3 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60F17 Functional limit theorems; invariance principles 60G51 Processes with independent increments; Lévy processes 60G52 Stable stochastic processes 60F05 Central limit and other weak theorems 60G44 Martingales with continuous parameter 60J65 Brownian motion 60J75 Jump processes (MSC2010) Keywords:non-linear Langevin type equation; stochastic differential equation; \(\alpha\)-stable Lévy process; Brownian motion; scaling limit; functional central limit theorem; martingales; exponential ergodic processes; Lyapunov function Citations:Zbl 1309.60059 PDFBibTeX XMLCite \textit{R. Eon} and \textit{M. Gradinaru}, Electron. J. Probab. 20, Paper No. 100, 19 p. (2015; Zbl 1328.60135) Full Text: DOI