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Variable order fractional Fokker–Planck equations derived from Continuous Time Random Walks

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  • Straka, Peter

Abstract

Continuous Time Random Walk models (CTRW) of anomalous diffusion are studied, where the anomalous exponent β(x)∈(0,1) varies in space. This type of situation occurs e.g. in biophysics, where the density of the intracellular matrix varies throughout a cell. Scaling limits of CTRWs are known to have probability distributions which solve fractional Fokker–Planck type equations (FFPE). This correspondence between stochastic processes and FFPE solutions has many useful extensions e.g. to nonlinear particle interactions and reactions, but has not yet been sufficiently developed for FFPEs of the “variable order” type with non-constant β(x).

Suggested Citation

  • Straka, Peter, 2018. "Variable order fractional Fokker–Planck equations derived from Continuous Time Random Walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 451-463.
    • Handle: RePEc:eee:phsmap:v:503:y:2018:i:c:p:451-463
    DOI: 10.1016/j.physa.2018.03.010
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    References listed on IDEAS

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    1. Erhan Cinlar, 1974. "Markov Additive Processes and Semi-Regeneration," Discussion Papers 118, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Sun, HongGuang & Chen, Wen & Chen, YangQuan, 2009. "Variable-order fractional differential operators in anomalous diffusion modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4586-4592.
    3. Straka, P. & Henry, B.I., 2011. "Lagging and leading coupled continuous time random walks, renewal times and their joint limits," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 324-336, February.
    4. Guido Germano & Mauro Politi & Enrico Scalas & Ren'e L. Schilling, 2008. "Stochastic calculus for uncoupled continuous-time random walks," Papers 0802.3769, arXiv.org, revised Jan 2009.
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    Cited by:

    1. Hidekazu Yoshioka & Kunihiko Hamagami & Haruka Tomobe, 2023. "A Non-local Fokker-Planck Equation with Application to Probabilistic Evaluation of Sediment Replenishment Projects," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-37, March.
    2. Sibatov, Renat T. & L'vov, Pavel E. & Sun, HongGuang, 2024. "Variable-order fractional diffusion: Physical interpretation and simulation within the multiple trapping model," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    3. Sergei Fedotov & Alexey O. Ivanov & Hong Zhang, 2025. "Anomalous Transport of Heterogeneous Population and Time-Changed Pólya Process," Mathematics, MDPI, vol. 13(12), pages 1-12, June.

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