L\'evy flights on the half line

TL;DR

This paper derives an explicit analytical approximation for the probability distribution of a Le9vy flight near an absorbing boundary, using perturbation methods around Brownian motion and confirming results with simulations.

Contribution

It introduces a first-order perturbation solution for the Le9vy flight distribution near an absorbing wall, extending understanding beyond the Brownian case.

Findings

Explicit first-order analytical solution for the pdf

Conjectured asymptotic behavior for all e4

Numerical and exact validation for e4=1

Abstract

We study the probability distribution function (pdf) of the position of a L\'evy flight of index 0<\alpha<2 in presence of an absorbing wall at the origin. The solution of the associated fractional Fokker-Planck equation can be constructed using a perturbation scheme around the Brownian solution (corresponding to \alpha = 2), as an expansion in \epsilon = 2 - \alpha. We obtain an explicit analytical solution, exact at the first order in \epsilon, which allows us to conjecture the precise asymptotic behavior of this pdf, including the first subleading corrections, for any \alpha. Careful numerical simulations, as well as an exact computation for \alpha = 1, confirm our conjecture.