Abstract
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 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 Hintze and Pavlyukevich.<br />
Highlights
In this paper we consider the one-dimensional and non-linear Langevin type equation driven by a symmetric α-stable Lévy process
Let us denote by xεt the one-dimensional process describing the position of a particle at time t ≥ 0, having the speed vtε t xεt = x0 + vsεds, t ≥ 0, (1.1)
In the paper by Hintze and Pavlyukevich, the authors study the asymptotic behaviour of the integrated Ornstein-Uhlenbeck and prove that this process converges weakly, as ε → 0, to the underlying α-stable Lévy process
Summary
In this paper we consider the one-dimensional and non-linear Langevin type equation driven by a symmetric α-stable Lévy process. In the paper by Hintze and Pavlyukevich, the authors study the asymptotic behaviour of the integrated Ornstein-Uhlenbeck and prove that this process converges weakly, as ε → 0, to the underlying α-stable Lévy process. Our goal is to answer the same question in the situation of a super-harmonic potential: what is the asymptotic behaviour of the position process xεt , as ε → 0 ? To get convergence toward a stable process, one needs to change the approach and other technical difficulties appear This case would be presented in a forthcoming work (see [8])
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