Continuous time random walk and parametric subordination in fractional diffusion

Abstract

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.

28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'. Denton, Texas, August 2005

References in corpus (5)

• Fractional Calculus: Integral and Differential Equations of Fractional Order
• The fundamental solution of the space-time fractional diffusion equation
• Mellin transform and subordination laws in fractional diffusion processes
• From Power Laws to Fractional Diffusion: the Direct Way
• Black-Scholes model under subordination

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