Mittag-Leffler Waiting Time, Power Laws,Rarefaction, Continuous Time Random Walk, Diffusion Limit

TL;DR

This paper explores the use of Mittag-Leffler functions in renewal processes and continuous time random walks, demonstrating their asymptotic equivalence to power laws and deriving diffusion limits for fractional processes.

Contribution

It establishes the asymptotic equivalence between power law and Mittag-Leffler waiting times and derives diffusion limits for space-time fractional processes using rescaling techniques.

Findings

Mittag-Leffler waiting times are asymptotically equivalent to power law waiting times.

Diffusion limits of CTRWs under power law regimes are obtained via rescaling.

Time-fractional drift processes are shown as limits of fractional Poisson processes.

Abstract

We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic power law waiting time to the Mittag-Leffler waiting time distribution via rescaling and respeeding the clock of time. By a second respeeding (by rescaling the spatial variable) we obtain the diffusion limit of the continuous time random walk under power law regimes in time and in space. Finally, we exhibit the time-fractional drift process as a diffusion limit of the fractional Poisson process and as a subordinator for space-time fractional diffusion.