The Spectrum of the Fractional Laplacian and First Passage Time Statistics

TL;DR

This paper derives exact spectral results for the fractional Laplacian in bounded domains and demonstrates that the full distribution, not just the average, is essential for understanding First Passage Time statistics of Lévy flights.

Contribution

It provides a novel exact spectral analysis of the fractional Laplacian and introduces an efficient method to compute the entire FPT distribution beyond the average.

Findings

FPT distribution is not centered around the mean

Knowledge of the full distribution is crucial for accurate FPT analysis

New method enables calculation of higher order cumulants

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 L\'evy 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.