First passage and first hitting times of Lévy flights and Lévy walks

Vladimir V Palyulin,Michael A Lomholt,Rainer Klages,George Blackburn,Nicholas W Watkins,Ralf Metzler,Aleksei V Chechkin

doi:10.1088/1367-2630/ab41bb

Abstract

For both Lévy flight and Lévy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Lévy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it (‘leapovers’), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the Lévy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.

Highlights

When a stochastic process x(t) first reaches a given threshold value in many scenarios follow-up events are triggered: shares are sold when their value crosses a pre-set target amount, or chemical reactions occur when two reactive particles encounter each other in space
The probability density function (PDF) for the relocation times for Lévy walk (LW) follows from the jump length PDF of Lévy flight (LF), as we show in appendix B
We demonstrated that for α > 1 the results of these two models are qualitatively identical at long times due to the finite average length of a relocation

Summary

Introduction

When a stochastic process x(t) first reaches a given threshold value in many scenarios follow-up events are triggered: shares are sold when their value crosses a pre-set target amount, or chemical reactions occur when two reactive particles encounter each other in space. While for Brownian motion the events of first-passage and first-hitting (or first-arrival) are identical because space is being explored continuously [54], the possibility of long, non-local jumps lead to ‘leapovers’ [55, 56], single jumps in which a given point is overshot by some leapover length, as illustrated in figure 2: for random walk processes with diverging variance of the jump length PDF, the event of first-passage becomes fundamentally different from that of first-hitting, and it is intuitively clear that first-hitting a target is harder (less likely) than the first-passage. We here systematically investigate the first-passage and firsthitting properties of LFs and LWs in one dimension by drawing generic conclusions on their differences and similarities This is an important first step in the assessment of these two fundamental random search processes. A summary and discussion is provided in section 6, and details of the mathematical derivations are deferred to the appendices

The role of Lévy flights and walks in random target search

Setup of the system: determining first-passage and first-hitting times

First-passage properties of Lévy flights and Lévy walks

First-hitting properties of Lévy flights and Lévy walks

Conclusions