Mean first passage time and absorption probabilities of a Lévy flier on a finite interval: discrete space and continuous limit via Fock space approach

Abstract

We have extended the Fock space (FS) approach to address the relevant problem of evaluating the mean first passage time (MFPT) and absorption probabilities of a Lévy random particle in a discrete finite interval with absorbing boundaries. In this approach the master equation is cast in the form of a real-valued Schrödinger-like equation with a Hamiltonian-like operator related to the transition probabilities, defined by the Lévy jump lengths. We present a comprehensive study of such quantities as a function of the eigenvalues and eigenvectors of the quasi-Hamiltonian, Lévy stability index α, and the particle’s starting position. The predicted asymptotic behavior of the absorption probabilities is finely captured under the FS approach. Moreover, by considering as well the continuous space limit, our findings nicely match the exact analytical result for a Lévy flier in continuous bounded space. For comparison, we also provide Monte Carlo results with good agreement with the FS calculations. The MFPT and absorption probabilities are relevant in a number of practical contexts, such as animal foraging and light transmission in randomly scattering media, and our findings may be useful to advance the understanding of these systems.