First passage leapovers of Lévy flights and the proper formulation of absorbing boundary conditions

Abstract

An important open problem in the theory of Lévy flights concerns the analytically tractable formulation of absorbing boundary conditions. Although the nonlocal approach, where the dynamical operators are given restrictions on their domain properties, has yielded substantial insights regarding the statistics of first passage, the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these nonlocal restrictions. In this study it is demonstrated that using the first-hit distribution, related to the first passage leapover, as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing Lévy flight. In particular, knowledge of the first-hit distribution is sufficient to fully determine the first passage time and position density of the particle, without requiring operator domain restrictions or numerical simulations. In addition, we report on the first-hit and leapover properties of first passages and arrivals for Lévy flights of arbitrary skew parameter, and extend these results to Lévy flights in a certain ubiquitous class of potentials satisfying an integral condition.