  For both Levy flight and Levy 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 Levy 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 Levy 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.
