Issue
EPL
Volume 83, Number 3, August 2008
Article Number 30006
Number of page(s) 6
Section General
DOI http://dx.doi.org/10.1209/0295-5075/83/30006
Published online 17 July 2008
EPL, 83 (2008) 30006
DOI: 10.1209/0295-5075/83/30006

The spectrum of the fractional Laplacian and First-Passage$\hbox{--}$Time statistics

E. Katzav and M. Adda-Bedia

Laboratoire de Physique Statistique de l'Ecole Normale Supérieure, CNRS UMR 8550 24 rue Lhomond, 75231 Paris Cedex 05, France, EU

eytan.katzav@lps.ens.fr

received 18 April 2008; accepted in final form 9 June 2008; published August 2008
published online 17 July 2008

Abstract
We present exact results for the spectrum of the fractional Laplacian in a bounded domain and apply them to First-Passage$\hbox{--}$Time (FPT) statistics of Lévy flights. We specifically show that the average is insufficient to describe the distribution of the FPT, although it is the only quantity available in the existing literature. In particular, we show that the FPT distribution is not peaked around the average, and that knowledge of the whole distribution is necessary to describe this phenomenon. For this purpose, we provide an efficient method to calculate higher-order cumulants and the whole distribution.

PACS
05.40.Fb - Random walks and Lévy flights.
02.50.-r - Probability theory, stochastic processes, and statistics.
89.65.Gh - Economics; econophysics, financial markets, business and management.

© EPLA 2008