Lévy flights on the half line

Abstract

We study the probability distribution function (pdf) of the position of a Lévy flight of index 0<α<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 α= 2), as an expansion in ε= 2 - α. We obtain an explicit analytical solution, exact at the first order in ε, which allows us to conjecture the precise asymptotic behavior of this pdf, including the first subleading corrections, for any α. Careful numerical simulations, as well as an exact computation for α= 1, confirm our conjecture.

11 pages, 9 figures. Revised version

References in corpus (5)

• Fractional Laplacian in Bounded Domains
• Asymptotic behavior of self-affine processes in semi-infinite domains
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• Vicious Lévy flights
• Area distribution and the average shape of a Lévy bridge

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