Continuous-time random walk and parametric subordination in fractional diffusion

By Rudolf Gorenflo, Francesco Mainardi, Alessandro Vivoli. 2007

173 citations recorded by the source corpus.

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 Levy process, we generate and display sample paths for some special cases.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/555beaca2404d0c6fbe06849a55face4d4354441

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In Applied Mathematics

1. Kei Kobayashi
2. Sabir Umarov
3. S. Agrawal
4. M. Srivastava
5. Subir Das
6. Marjorie G. Hahn

In Modeling and Simulation

1. Sabir Umarov
2. S. Agrawal
3. M. Srivastava
4. Subir Das
5. Marjorie G. Hahn
6. Kei Kobayashi

Citations in the corpus

• Synchronization of fractional order chaotic systems using active control method (2012, background): “[21] demonstrated that for obtaining the space–time fractional diffusion equation describing diffusion process, 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 deriv”
• SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations (2009, background): “A fractional generalization of the Cauchy problem (1.1) with A = L(x, Dx) ,i n the sense that the first-order time derivative on the left side of (1.1) is replaced by a time-fractional order derivative, has appeared in the framework of continuous-time random walks (CTRWs) and fractional kinetic theo”