Thin and Thick Strip Passage Times for Lévy Flights and Lévy Processes

Abstract: We review some of the theory relevant to passage times of one-dimensional L\'evy processes out of bounded regions, highlighting results that are useful in physical phenomena modelled by heavy-tailed L\'evy flights. The process is hypothesised to describe the motion of a particle on the line, starting at $0$, and exiting either a fixed interval $[-r, r]$, $r > 0$, or a time-dependent, expanding, set of intervals of the form $[-r t^\kappa, r t^\kappa]$, $r > 0$, $\kappa > 0$. Asymptotic behaviour of the exit time may be as $r \downarrow 0$ or as $r \to \infty$, but particular emphasis is placed herein on "small time" approximations, corresponding to exits from or transmissions through thin strips. Applications occur for example in the transmission of photons through moderately doped thin or thick wafers by means of "photon recycling", and in atmospheric radiation modelling.