The Spectrum of the Fractional Laplacian and First Passage Time Statistics

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 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.

arXiv comment10 pages, 6 figures

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• First passage time moments of asymmetric Lévy flights
• First passage times for subordinate Brownian motions
• Seeing asymptotic freedom in an exact correlator of a large-$N$ matrix field theory
• The universal coefficient of the exact correlator of a large-$N$ matrix field theory
• Role of long jumps in Lévy noise induced multimodality
• Deterministic force-free resonant activation