# Trajectory statistics of confined L\\'{e}vy flights and Boltzmann-type equilibria

Source: [Trajectory statistics of confined L\\'{e}vy flights and Boltzmann-type equilibria](<https://www.sota2.com/research/paper/68fa30cb1e209131ce878a4e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:53:42.005Z

Authors: Mariusz Zaba, Piotr Garbaczewski, Vladimir Stephanovich

Published: 2013-03-27T18:24:57.000Z

## Abstract

We analyze a specific class of random systems that are driven by a symmetric L\\'{e}vy stable noise, where Langevin representation is absent. In view of the L\\'{e}vy noise sensitivity 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 \[-\\Phi (x)\]$. Here, we infer pdf $\\rho (x,t)$ based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for $\\rho (x,t)$ and its target pdf $\\rho\_\*(x)$. To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems where it can be useful.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1303.6920>)

[PDF](<https://arxiv.org/pdf/1303.6920.pdf>)