Alpha-stable Levy motion stands for a class of non-Gaussian Markovian random processes whose stationary increments are distributed according to the Levy stable probability laws. Such process serves as a limit continuous-time description of Levy flight (LF), a discrete-time random walk whose jumps or flights exhibit a scale-free heavy-tailed distribution with infinite variance. It is generally believed that Levy flights optimize random search for sparse targets. In the present talk we give a short review of temporal and spatial properties of alpha-stable motion, focusing on the distinction between the first passage (or first escape) and first arrival phenomena. Based on this probabilistic background, we study the efficiency of several minimalist search models based on pure and combined Levy flight search strategies, in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space). We deduce certain common patterns from these models.

* V.V. Palyulin, V.N. Mantsevich, R. Klages, R. Metzler, and A.V. Chechkin, Comparison of pure and combined search strategies for single and multiple targets. Eur. Phys. J. B (2017, Special Issue, accepted).
* B. Dybiec, E. Gudowska-Nowak, A.V. Chechkin, To hit or to pass it over—remarkable transient behavior of first arrivals and passages for Levy flights in finite domains J. Phys. A: Math. Theor. 49 (50), 504001 (2016).
* V.V. Palyulin, A.V. Chechkin, R. Klages, R. Metzler, Search reliability and search efficiency of combined Levy–Brownian motion: long relocations mingled with thorough local exploration. J. Phys. A: Math. Theor. 49 (39), 394002 (2016).
* V.V. Palyulin, A.V. Chechkin, R. Metzler, Space-fractional Fokker-Planck equation and optimization of random search processes in the presence of an external bias. J. Stat. Mech., P11031 (2014)
* V.V. Palyulin, A.V. Chechkin, R. Metzler, Levy flights do not always optimize random blind search for sparse targets. Proc. Natl. Acad. Sci. USA 111, No.8, 2931-2936 (2014).
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* T. Koren, M.A. Lomholt, A.V. Chechkin, J. Klafter, R. Metzler, Leapover lengths and first passage time statistics for Levy flights, Phys. Rev. Lett. 99, 160602 (2007).
* R. Metzler, A.V. Chechkin, V.Yu. Gonchar, J. Klafter, Some fundamental aspects of Levy flights, Chaos, Solitons Fractals 34, 129 (2007).
* T. Koren, A.V. Chechkin, J. Klafter, On the first passage time and leapover properties of Levy motions, Physica A 379, 10 (2007).
* A.V. Chechkin, R. Metzler, V.Yu. Gonchar, J. Klafter, L.V. Tanatarov. First passage and arrival time densities for Levy flights and the failure of the method of images. J. Phys. A: Math. Gen. 36, L537 (2003).