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

摘要

We derive a functional equation for the mean first-passage time (MFPT) of ageneric self-similar Markovian continuous process to a target in aone-dimensional domain and obtain its exact solution. We show that the obtainedexpression of the MFPT for continuous processes is actually different from thelarge system size limit of the MFPT for discrete jump processes allowingleapovers. In the case considered here, the asymptotic MFPT admitsnon-vanishing corrections, which we call residual MFPT. The case of L/'evyflights with diverging variance of jump lengths is investigated in detail, inparticular, with respect to the associated leapover behaviour. We also shownumerically that our results apply with good accuracy to fractional Brownianmotion, despite its non-Markovian nature.