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August 2008 EPL, 83 (2008) 30006 www.epljournal.org

Summary: August 2008
EPL, 83 (2008) 30006 www.epljournal.org
doi: 10.1209/0295-5075/83/30006
The spectrum of the fractional Laplacian and First-Passage­Time
E. Katzav(a)
and M. Adda-Bedia
Laboratoire de Physique Statistique de l'Ecole Normale Sup´erieure, CNRS UMR 8550
24 rue Lhomond, 75231 Paris Cedex 05, France, EU
received 18 April 2008; accepted in final form 9 June 2008
published online 17 July 2008
PACS 05.40.Fb ­ Random walks and L´evy flights
PACS 02.50.-r ­ Probability theory, stochastic processes, and statistics
PACS 89.65.Gh ­ Economics; econophysics, financial markets, business and management
Abstract ­ We present exact results for the spectrum of the fractional Laplacian in a bounded
domain and apply them to First-Passage­Time (FPT) statistics of L´evy 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-


Source: Adda-Bedia, Mokhtar - Laboratoire de Physique Statistique, Département de Physique, École Normale Supérieure


Collections: Physics