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Zbl 1142.82363
Gorenflo, Rudolf; Mainardi, Francesco; Vivoli, Alessandro
Continuous-time random walk and parametric subordination in fractional diffusion. (English)
[J] Chaos Solitons Fractals 34, No. 1, 87-103 (2007). ISSN 0960-0779

Summary: The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit, we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Lévy process, we generate and display sample paths for some special cases.

MSC 2000:: *82C41 Dynamics of random walks, etc.
82C70 Transport processes

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