Asymmetric Lévy flights in the presence of absorbing boundaries

Clélia de Mulatier 1, 2, Alberto Rosso 1, Gregory Schehr 1

Journal of Statistical Mechanics: Theory and Experiment (2013) P10006

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 Lévy 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 Lévy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Département de Modélisation des Systèmes et Structures (DM2S),
    CEA : DEN/DM2S