  An important open problem in the theory of L\'evy flights concerns the
analytically tractable formulation of absorbing boundary conditions. Although
numerical studies using the correctly defined nonlocal approach have 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 conditions. 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 L\'evy 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 integral truncation or
numerical simulations. In addition, we report on the first-hit and leapover
properties of first passages and arrivals for L\'evy flights of arbitrary skew
parameter, and extend these results to L\'evy flights in a certain ubiquitous
class of potentials satisfying an integral condition.

Statistical Mechanics