Harmonic oscillator under Levy noise: Unexpected properties in the phase space

TL;DR

This paper investigates the effects of Levy noise on a harmonic oscillator, revealing unexpected non-Gaussian properties such as inhomogeneous phase distribution and strong dependence between position and velocity.

Contribution

It demonstrates how Levy noise fundamentally alters the phase space properties of the harmonic oscillator, contrasting with the well-understood Gaussian case.

Findings

Level lines of joint probability density are not elliptic under Levy noise

Position and velocity are strongly dependent in Levy noise case

Phase distribution becomes inhomogeneous and complex

Abstract

A harmonic oscillator under influence of the noise is a basic model of various physical phenomena. Under Gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate Gaussian distribution with elliptic level lines. The distribution of phase is homogeneous. None of these properties hold in the general L\'evy case. Thus, the level lines of the joint probability density are not elliptic. The coordinate and the velocity of the oscillator are strongly dependent, and this dependence is quantified by introducing the corresponding parameter ("width deficit"). The distribution of the phase is inhomogeneous and highly nontrivial.