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

2010/04/26 by Rudolf Gorenflo, Gorenflo, Rudolf
Biochemistry, Genetics and Molecular Biology · Mathematics · #26A33. 33E12 #45K05 #60G18 #60G50 #60G52 #60K05 #76R50. #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Probability (math.PR) #Statistical Distribution Estimation and Applications #Statistical Mechanics (cond-mat.stat-mech)

paper · pdf · doi:10.48550/arxiv.1004.4413

openalex publication_date 2010/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Continuous-time random walk and parametric subordination in fractional diffusion
• Fractional Poisson Process
• Fractional Differential Equations
• Power laws, Pareto distributions and Zipf's law

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