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Fokker-Planck and Kolmogorov backward equations for continuous time random walk scaling limits


Authors: Boris Baeumer and Peter Straka
Journal: Proc. Amer. Math. Soc. 145 (2017), 399-412
MSC (2010): Primary 60F17; Secondary 60G22
DOI: https://doi.org/10.1090/proc/13203
Published electronically: July 12, 2016
MathSciNet review: 3565391
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Abstract: It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.


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Additional Information

Boris Baeumer
Affiliation: Department of Mathematics & Statistics, University of Otago, North Dunedin, Dunedin 9016, New Zealand
Email: bbaeumer@maths.otago.ac.nz

Peter Straka
Affiliation: School of Mathematics and Statistics, University of New South Wales Australia, Sydney, NSW 2052, Australia
Email: p.straka@unsw.edu.au

DOI: https://doi.org/10.1090/proc/13203
Keywords: Anomalous diffusion, fractional kinetics, fractional derivative, subordination, coupled random walks
Received by editor(s): December 12, 2014
Received by editor(s) in revised form: February 26, 2016, and March 16, 2016
Published electronically: July 12, 2016
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society