Mittag-Leffler Waiting Time, Power Laws,Rarefaction, Continuous Time Random Walk, Diffusion Limit
arXiv:1004.4413
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.
28 pages. Invited Lecture, Workshop on Fractional Calculus and Statistical Distributions, November 25-27, 2009, Centre for Mathematical Sciences, Pala Campus, Kerala-India. To appear in the Proceedings.
References in corpus (5)
- Power laws, Pareto distributions and Zipf's law
- Fractional Calculus: Integral and Differential Equations of Fractional Order
- The fundamental solution of the space-time fractional diffusion equation
- Continuous time random walk and parametric subordination in fractional diffusion
- From Power Laws to Fractional Diffusion: the Direct Way
Cited by in corpus (8)
- Subordination Pathways to Fractional Diffusion
- Continuous time random walk and diffusion with generalized fractional Poisson process
- On Discrete Time Prabhakar-Generalized Fractional Poisson Processes and Related Stochastic Dynamics
- On the fractional Poisson process and the discretized stable subordinator
- Biased continuous-time random walks with Mittag-Leffler jumps
- Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations
- Asymmetric random walks with bias generated by discrete-time counting processes
- Controlled Continuous Time Random Walks and fractional Hamilton Jacobi Bellman equations