Abstract Chechkin: From first passage and arrival for alpha-stable random motion to efficiency and reliability for Levy flight random search

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).

* R. Metzler, A.V. Chechkin, J. Klafter, Levy Statistics and Anomalous Transport: Levy Flights and Subdiffusion. In: Encyclopedia of Complexity and System Science, edited by R. Mayers. Springer Science + Business Media, LLC, New York, 2009, pp.1724-1745.

* 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).

* 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).

* R. Metzler, A.V. Chechkin, J. Klafter, Levy Statistics and Anomalous Transport: Levy Flights and Subdiffusion. In: Encyclopedia of Complexity and System Science, edited by R. Mayers. Springer Science + Business Media, LLC, New York, 2009, pp.1724-1745.

* 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).