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Continuous-time random walk and parametric subordination in fractional diffusion. (English) Zbl 1142.82363
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:
82C41Dynamics of random walks, random surfaces, lattice animals, etc.
82C70Transport processes (time-dependent statistical mechanics)