Asymmetric L\'evy 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) c/ηα +1 for large positive jumps and φ(η) 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, xn, converges to an asymmetric L\'evy stable law of stability index α and skewness parameter β=(γ-1)/(γ+1). In particular the right tail of this PDF decays as c n/xn1+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+) \, xn1+α]. 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.