From Power Laws to Fractional Diffusion: the Direct Way

Abstract

Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that, by a well-scaled passage to the diffusion limit, generalized diffusion processes, fractional in time as well as in space, are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure.

12 pages, 4 figures

References in corpus (2)

• Fractional Calculus: Integral and Differential Equations of Fractional Order
• The fundamental solution of the space-time fractional diffusion equation

Cited by in corpus (5)

• Continuous time random walk and parametric subordination in fractional diffusion
• A fractional generalization of the Poisson processes
• Mittag-Leffler Waiting Time, Power Laws,Rarefaction, Continuous Time Random Walk, Diffusion Limit
• Continuous time random walk and diffusion with generalized fractional Poisson process
• Some recent advances in theory and simulation of fractional diffusion processes