Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations

Article


Angstmann, C. N., Donnelly, I. C., Henry, B. I., Langlands, T. A. M. and Straka, P.. 2015. "Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations." SIAM Journal on Applied Mathematics. 75 (4), pp. 1445-1468. https://doi.org/10.1137/15M1011299
Article Title

Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations

ERA Journal ID393
Article CategoryArticle
AuthorsAngstmann, C. N. (Author), Donnelly, I. C. (Author), Henry, B. I. (Author), Langlands, T. A. M. (Author) and Straka, P. (Author)
Journal TitleSIAM Journal on Applied Mathematics
Journal Citation75 (4), pp. 1445-1468
Number of Pages24
Year2015
Place of PublicationUnited States
ISSN0036-1399
1095-712X
Digital Object Identifier (DOI)https://doi.org/10.1137/15M1011299
Web Address (URL)http://epubs.siam.org/doi/abs/10.1137/15M1011299
Abstract

Continuous time random walks, which generalize random walks by adding a stochastic time between jumps, provide a useful description of stochastic transport at mesoscopic scales. The continuous time random walk model can accommodate certain features, such as trapping, which are not manifest in the standard macroscopic diffusion equation. The trapping is incorporated through a waiting time density, and a fractional diffusion equation results from a power law waiting time. A generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force. Here we have derived the master equations for continuous time random walks with space- and time-dependent forcing for two cases: when the force is evaluated at the start of the waiting time and at the end of the waiting time. The differences
persist in low order spatial continuum approximations; however, the two processes are shown to be governed by the same Fokker–Planck equations in the diffusion limit. Thus the fractional Fokker–Planck equation with space- and time-dependent forcing is robust to these changes in the underlying stochastic process.

Keywordscontinuous time random walk; fractional Fokker-Planck equation; anomalous diffusion; generalized master equation; limit theorems; fractional calculus
Contains Sensitive ContentDoes not contain sensitive content
ANZSRC Field of Research 2020490510. Stochastic analysis and modelling
490410. Partial differential equations
490199. Applied mathematics not elsewhere classified
Public Notes

File reproduced in accordance with the copyright policy of the publisher/author.

Byline AffiliationsUniversity of New South Wales
Department of Mathematics and Computing
Institution of OriginUniversity of Southern Queensland
Funding source
Australian Research Council (ARC)
Grant ID
DP140101193
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