First passage and arrival time densities for L\'evy flights and the failure of the method of images

TL;DR

This paper investigates the first passage time problem for Lévy flights, revealing that the traditional method of images fails and that the first arrival density differs from the first passage time density, supported by numerical evidence.

Contribution

It demonstrates the failure of the method of images for Lévy flights and clarifies the distinction between first arrival and first passage time densities.

Findings

Method of images violates Sparre Andersen's theorem for Lévy flights.

First arrival density differs from first passage time density in Lévy flights.

Numerical results support the theoretical findings.

Abstract

We discuss the first passage time problem in the semi-infinite interval, for homogeneous stochastic Markov processes with L{\'e}vy stable jump length distributions $\lambda(x)\sim\ell^{\alpha}/|x|^{1+\alpha}$ ($|x|\gg\ell$), namely, L{\'e}vy flights (LFs). In particular, we demonstrate that the method of images leads to a result, which violates a theorem due to Sparre Andersen, according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density (FPTD) governed by the universal long-time decay $\sim t^{-3/2}$. Conversely, we show that for LFs the direct definition known from Gaussian processes in fact defines the probability density of first arrival, which for LFs differs from the FPTD. Our findings are corroborated by numerical results.