TY  - JOUR
A1  - Padash, Amin
A1  - Chechkin, Aleksei V.
A1  - Dybiec, Bartlomiej
A1  - Pavlyukevich, Ilya
A1  - Shokri, Babak
A1  - Metzler, Ralf
T1  - First-passage properties of asymmetric Levy flights
JF  - Journal of physics : A, Mathematical and theoretical
N2  - Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index  and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.
KW  - Levy flights
KW  - first-passage
KW  - search dynamics
Y1  - 2019
U6  - https://doi.org/10.1088/1751-8121/ab493e
SN  - 1751-8113
SN  - 1751-8121
VL  - 52
IS  - 45
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Palyulin, Vladimir V.
A1  - Blackburn, George
A1  - Lomholt, Michael A.
A1  - Watkins, Nicholas W.
A1  - Metzler, Ralf
A1  - Klages, Rainer
A1  - Chechkin, Aleksei V.
T1  - First passage and first hitting times of Levy flights and Levy walks
JF  - New journal of physics : the open-access journal for physics
N2  - 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.
KW  - Levy flights
KW  - Levy walks
KW  - first-passage time
KW  - first-hitting time
Y1  - 2019
U6  - https://doi.org/10.1088/1367-2630/ab41bb
SN  - 1367-2630
VL  - 21
IS  - 10
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  -