Levy flights in confining environments: Random paths and their statistics

TL;DR

This paper investigates the behavior of Levy flights in confining environments, focusing on path-wise descriptions and statistical properties of jump trajectories driven by symmetric Levy stable noise, using numerical methods including a modified Gillespie algorithm.

Contribution

It establishes a path-wise framework for Levy flights in confining potentials and introduces a numerical approach to analyze jump path statistics without a Langevin representation.

Findings

Sample trajectories show predominance of small or large jumps depending on stability index

The method captures qualitative features of jump paths in Levy flights

Numerical simulations align with theoretical equilibrium distributions

Abstract

We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise. In view of the L\'{e}vy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental inhomogeneities), the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Since there is no Langevin representation of the dynamics in question, our main goal here is to establish the appropriate path-wise description of the underlying jump-type process and next infer the $\rho (x,t)$ dynamics directly from the random paths statistics. A priori given data are jump transition rates entering the master equation for $\rho (x,t)$ and its target pdf $\rho_*(x)$. We use numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices $\mu \in (0,2)$.