Average time spent by Lévy flights and walks on an interval with absorbing boundaries

S. V. Buldyrev, S. Havlin, A. Ya Kazakov, M. G.E. da Luz, E. P. Raposo, H. E. Stanley, G. M. Viswanathan

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We consider a Lévy flyer of order [formula presented] that starts from a point [formula presented] on an interval [formula presented] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of [formula presented] and the cases of [formula presented] and [formula presented] For [formula presented] when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if [formula presented] is placed in the vicinity of absorbing boundaries, the average total length has a minimum at [formula presented] corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

Original languageEnglish
Pages (from-to)11
Number of pages1
JournalPhysical Review E
Issue number4
StatePublished - 2001


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