Asymmetric Lévy flights in the presence of absorbing boundaries

Abstract

We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution ϕ(η) displaying asymmetric power law tails (i.e. ϕ(η) \sim c/η^{α+1} for large positive jumps and ϕ(η) \sim c/(γ|η|^{α+1}) for large negative jumps, with 0 < α< 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 Lévy stable law of stability index αand skewness parameter β=(γ-1)/(γ+1). In particular the right tail of this PDF decays as c n/x_n^{1+α}. 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 θ_+, 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-θ_+) \, x_n^{1+α}]. This last result can be generalized in higher dimensions such as a planar Lévy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.