The spectrum of the fractional Laplacian and First-Passage–Time statistics

2008/02/29 by Eytan Katzav, E. Katzav, Mokhtar Adda-Bedia +1
Decision Sciences · Mathematics · Physics and Astronomy · #Fractional Differential Equations Solutions #Probabilistic and Robust Engineering Design #cond-mat.dis-nn #cond-mat.soft #cond-mat.stat-mech #stochastic dynamics and bifurcation

paper · pdf · doi:10.1209/0295-5075/83/30006

published as Europhys. Lett 83 30006 (2008) · 10 pages, 6 figures

arxiv created 2008/05/28 · openalex publication_date 2008/07/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We present exact results for the spectrum of the fractional Laplacian in a bounded domain and apply them to First-Passage–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.

Citations

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