Stationary States in Bistable System Driven by L\'evy Noise

TL;DR

This paper investigates the stationary probability density functions of a bistable system influenced by heavy-tailed Le9vy noise, revealing differences from Gaussian noise and analyzing the effects of noise parameters.

Contribution

It provides an analytical solution for the stationary PDF at e1 = 1 and employs numerical methods for arbitrary e1, highlighting how Le9vy noise alters the PDF's maxima positions.

Findings

Stationary PDF shape differs from Gaussian noise case.

Maxima of the PDF do not align with potential minima under Le9vy noise.

The distance between maxima and minima depends on noise parameters.

Abstract

We study the properties of the probability density function (PDF) of a bistable system driven by heavy tailed white symmetric L\'evy noise. The shape of the stationary PDF is found analytically for the particular case of the L\'evy index \alpha = 1 (Cauchy noise). For an arbitrary L\'evy index we employ numerical methods based on the solution of the stochastic Langevin equation and space fractional kinetic equation. In contrast with the bistable system driven by Gaussian noise, in the L\'evy case the positions of maxima of the stationary PDF do not coincide with the positions of minima of the bistable potential. We provide a detailed study of the distance between the maxima and the minima as a function of the potential's depth and L\'evy noise parameters.