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For both Levy flight and Levy walk search processes we analyse the fulldistribution of first-passage and first-hitting (or first-arrival) times. Theseare, respectively, the times when the particle moves across a point at somegiven distance from its initial position for the first time, or when it landsat a given point for the first time. For Levy motions with their propensity forlong relocation events and thus the possibility to jump across a given point inspace without actually hitting it ("leapovers"), these two definitions lead tosignificantly different results. We study the first-passage and first-hittingtime distributions as functions of the Levy stable index, highlighting thedifferent behaviour for the cases when the first absolute moment of the jumplength distribution is finite or infinite. In particular we examine the limitsof short and long times. Our results will find their application in themathematical modelling of random search processes as well as computeralgorithms.

First-passage and first-hitting times of Levy flights and Levy walks