Thin and Thick Strip Passage Times for L\'evy Flights and L\'evy Processes

TL;DR

This paper reviews theoretical results on the passage times of one-dimensional Lévy processes, focusing on small-time behavior relevant to physical phenomena like photon transmission and atmospheric radiation.

Contribution

It provides a comprehensive review of passage time theory for Lévy processes, emphasizing applications to thin and thick strip transmission in physical models.

Findings

Asymptotic behavior analyzed for small and large regions.

Focus on small-time approximations for thin strips.

Applications in photon recycling and atmospheric radiation.

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.