Residual mean first-passage time for jump processes: theory and applications to L\'evy flights and fractional Brownian motion

TL;DR

This paper derives an exact functional equation for the mean first-passage time of self-similar Markovian processes, revealing residual effects in jump processes like Lévy flights and extending insights to fractional Brownian motion.

Contribution

It introduces a novel functional equation for MFPT in self-similar processes and highlights the residual MFPT in jump processes, with applications to Lévy flights and fractional Brownian motion.

Findings

Exact solution for MFPT of self-similar Markovian processes.

Identification of residual MFPT corrections in jump processes.

Numerical validation for fractional Brownian motion.

Abstract

We derive a functional equation for the mean first-passage time (MFPT) of a generic self-similar Markovian continuous process to a target in a one-dimensional domain and obtain its exact solution. We show that the obtained expression of the MFPT for continuous processes is actually different from the large system size limit of the MFPT for discrete jump processes allowing leapovers. In the case considered here, the asymptotic MFPT admits non-vanishing corrections, which we call residual MFPT. The case of L/'evy flights with diverging variance of jump lengths is investigated in detail, in particular, with respect to the associated leapover behaviour. We also show numerically that our results apply with good accuracy to fractional Brownian motion, despite its non-Markovian nature.