@article{PalyulinChechkinMetzler2014,
  author    = {Palyulin, Vladimir V. and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Space-fractional Fokker-Planck equation and optimization of random search processes in the presence of an external bias},
  series = {Journal of statistical mechanics: theory and experiment},
  journal   = {Journal of statistical mechanics: theory and experiment},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1742-5468},
  doi       = {10.1088/1742-5468/2014/11/P11031},
  pages     = {32},
  year      = {2014},
  abstract  = {Based on the space-fractional Fokker-Planck equation with a delta-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index alpha = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for alpha may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.},
  language  = {en}
}
@article{MagdziarzMetzlerSzczotkaetal.2012,
  author    = {Magdziarz, Marcin and Metzler, Ralf and Szczotka, Wladyslaw and Zebrowski, Piotr},
  title     = {Correlated continuous-time random walks-scaling limits and Langevin picture},
  series = {Journal of statistical mechanics: theory and experiment},
  journal   = {Journal of statistical mechanics: theory and experiment},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1742-5468},
  doi       = {10.1088/1742-5468/2012/04/P04010},
  pages     = {18},
  year      = {2012},
  abstract  = {In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.},
  language  = {en}
}