Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects
Abstract
We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.
Cite
@article{arxiv.0705.0797,
title = {Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects},
author = {Rudolf Gorenflo and Francesco Mainardi},
journal= {arXiv preprint arXiv:0705.0797},
year = {2008}
}