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

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

    Research output: Contribution to journalArticlepeer-review

    39 Citations (Scopus)

    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.

    Original languageEnglish
    Article number103028
    JournalNew Journal of Physics
    Volume21
    Issue number10
    DOIs
    Publication statusPublished - 11 Oct 2019

    Keywords

    • first-hitting time
    • first-passage time
    • Lévy flights
    • Lévy walks

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