Asymmetric L\'evy flights in the presence of absorbing boundaries

TL;DR

This paper analyzes asymmetric Lévy flights with absorbing boundaries, deriving the tail behavior of the position distribution and the survival probability decay, with exact results for one-dimensional and higher-dimensional confined cases.

Contribution

It provides exact calculations of the tail decay and persistence exponents for asymmetric Lévy flights confined by absorbing boundaries, extending understanding of confined anomalous diffusion.

Findings

Tail of the position PDF decays as c n/[(1-θ_+) x^{1+α}]

Persistence exponent θ_+ is computed exactly for semi-infinite confinement

Results confirmed by numerical simulations

Abstract

We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution \phi(\eta) displaying asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large positive jumps and \phi(\eta) \sim c/(\gamma |\eta|^{\alpha +1}) for large negative jumps, with 0 < \alpha < 2). In absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, x_n, converges to an asymmetric L\'evy stable law of stability index \alpha and skewness parameter \beta=(\gamma-1)/(\gamma+1). In particular the right tail of this PDF decays as c n/x_n^{1+\alpha}. Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In this case, the persistence exponent \theta_+, which characterizes the algebraic large time decay of the survival probability, can be computed exactly and we show that the tail of the PDF of the walker position decays as c \, n/[(1-\theta_+) \, x_n^{1+\alpha}]. This last result can be generalized in higher dimensions such as a planar L\'evy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.