Levy--Brownian motion on finite intervals: Mean first passage time analysis

TL;DR

This paper analyzes the mean first passage time for a generalized Wiener process driven by Lévy stable noises on finite intervals, extending classical results to non-Gaussian noise scenarios and validating methods numerically.

Contribution

It provides a comprehensive analysis of first passage time statistics for Lévy-driven processes, including boundary condition setup and numerical validation, extending classical Gaussian results.

Findings

Derived formulas for mean first passage time under Lévy noise

Validated methods numerically against analytical results for Gaussian limit

Clarified boundary condition setup for non-Gaussian processes

Abstract

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by L\'evy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulae when the stability index $\alpha$ approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.