@article{PalyulinChechkinKlagesetal.2016,
  author    = {Palyulin, Vladimir V. and Chechkin, Aleksei V. and Klages, Rainer and Metzler, Ralf},
  title     = {Search reliability and search efficiency of combined Levy-Brownian motion: long relocations mingled with thorough local exploration},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {49},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8113/49/39/394002},
  pages     = {2189 -- 2193},
  year      = {2016},
  abstract  = {A combined dynamics consisting of Brownian motion and Levy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker-Planck equation combining unbiased Brownian motion and Levy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Levy flights with stable exponent alpha < 1, by itself implying zero probability of hitting a point on a line, leads to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent a of the Levy flight component.},
  language  = {en}
}
@article{PalyulinBlackburnLomholtetal.2019,
  author    = {Palyulin, Vladimir V. and Blackburn, George and Lomholt, Michael A. and Watkins, Nicholas W. and Metzler, Ralf and Klages, Rainer and Chechkin, Aleksei V.},
  title     = {First passage and first hitting times of Levy flights and Levy walks},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {21},
  journal   = {New journal of physics : the open-access journal for physics},
  number    = {10},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab41bb},
  pages     = {23},
  year      = {2019},
  abstract  = {For both L{\´e}vy flight and L{\´e}vy 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 L{\´e}vy 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 L{\´e}vy 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.},
  language  = {en}
}