Suppression of oscillations by Levy noise

TL;DR

This paper analytically demonstrates that Levy noise suppresses oscillations in stochastic systems, highlighting fundamental differences from Gaussian noise and resolving calculus ambiguities in stochastic calculus.

Contribution

It provides an exact steady-state solution for stochastic equations with Levy noise, revealing how Levy flights suppress oscillatory behavior and differ from Gaussian noise effects.

Findings

Levy noise suppresses quasi-periodical oscillations.

Lévy flights have smaller variations than Gaussian noise in the limit.

The difference in noise types resolves stochastic calculus ambiguities.

Abstract

We find analytical solution of pair of stochastic equations with arbitrary forces and multiplicative L\'evy noises in a steady-state nonequilibrium case. This solution shows that L\'evy flights suppress always a quasi-periodical motion related to the limit cycle. We prove that difference between stochastic systems driven by L\'evy and Gaussian noises is that the L\'evy variation $\Delta L\sim(\Delta t)^{1/\alpha}$ with the exponent $\alpha<2$ is much less than the Gaussian one $\Delta W\sim(\Delta t)^{1/2}$ in the $\Delta t\to 0$ limit. Moreover, this difference is shown to remove the problem of the calculus choice because related addition to the physical force is of order $(\Delta t)^{2/\alpha}\ll\Delta t$.