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

2015/04/24 by Ross Maller, Ross A. Maller, Maller, Ross A. +2
Biochemistry, Genetics and Molecular Biology · Decision Sciences · Mathematics · #60G50 #60G51 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics #math.PR #msc:60G50 #msc:60G51

paper · pdf · doi:10.48550/arxiv.1504.06400

arxiv created 2015/04/24 · openalex publication_date 2015/04/24 · arxiv updated 2015/04/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We review some of the theory relevant to passage times of one-dimensional Lévy processes out of bounded regions, highlighting results that are useful in physical phenomena modelled by heavy-tailed Lévy 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^κ, r t^κ], r > 0, κ> 0. Asymptotic behaviour of the exit time may be as r \downarrow 0 or as r → ∞, 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.

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