Stationary states in Langevin dynamics under asymmetric L\'evy noises

TL;DR

This paper explores the stationary probability densities of systems driven by asymmetric alpha-stable Levy noises, analyzing different potentials and comparing analytical and numerical solutions, while also addressing parameter estimation.

Contribution

It provides a comprehensive analysis of stationary states under asymmetric Levy noises, including methods for constructing solutions and estimating parameters, extending beyond Gaussian noise models.

Findings

Stationary densities depend on potential shape and noise asymmetry.

Analytical solutions are validated against numerical simulations.

Parameter estimation methods for stationary densities are developed.

Abstract

Properties of systems driven by white non-Gaussian noises can be very different from these systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by $\alpha$-stable L\'evy type noises, which provide natural extension to the Gaussian noise having however a new property mainly a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic and in generic double well potential models subjected to the action of $\alpha$-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.