Geodesic
A Semi-Markov Algorithm for Continuous Time Random Walk Limit Distributions
G. Gill1 ; P. Straka1
1 School of Mathematics & Statistics, UNSW Australia
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 34-50.

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The Semi-Markov property of Continuous Time Random Walks (CTRWs) and their limit processes is utilized, and the probability distributions of the bivariate Markov process (X(t),V(t)) are calculated: X(t) is a CTRW limit and V(t) a process tracking the age, i.e. the time since the last jump. For a given CTRW limit process X(t), a sequence of discrete CTRWs in discrete time is given which converges to X(t) (weakly in the Skorokhod topology). Master equations for the discrete CTRWs are implemented numerically, thus approximating the distribution of X(t). A consequence of the derived algorithm is that any distribution of initial age can be assumed as an initial condition for the CTRW limit dynamics. Four examples with different temporal scaling are discussed: subdiffusion, tempered subdiffusion, the fractal mobile/immobile model and the tempered fractal mobile/immobile model.
DOI : 10.1051/mmnp/201611303

G. Gill 1 ; P. Straka 1

1 School of Mathematics & Statistics, UNSW Australia 
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G. Gill; P. Straka. A Semi-Markov Algorithm for Continuous Time Random Walk Limit Distributions. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 34-50. doi : 10.1051/mmnp/201611303. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611303/

[1] Ralf Metzler, Joseph Klafter Phys. Rep. 1 77 2000

[2] Benjamin M. Regner, Dejan Vučinić, Cristina Domnisoru, Thomas M. Bartol, Martin W. Hetzer, Daniel M. Tartakovsky, Terrence J. Sejnowski Biophys. J. 1652 1660 2013

[3] Brian Berkowitz, Andrea Cortis, Marco Dentz, Harvey Scher Rev. Geophys. RG2003 2006

[4] Enrico Scalas Complex Netw. Econ. Interact. 3 16 2005

[5] Fidel Santamaria, Stefan Wils, Erik De Schutter, George J. Augustine Neuron 635 48 2006

[6] Daniel S. Banks, Cécile Fradin Biophys. J. 2960 71 2005

[7] Bruce I Henry, Trevor AM Langlands, Peter Straka. An introduction to fractional diffusion. In R L. Dewar and F Detering, editors, Complex Phys. Biophys. Econophysical Syst. World Sci. Lect. Notes Complex Syst., volume 9 of World Scientific Lecture Notes in Complex Systems, pages 37–90, Singapore, 2010. World Scientific.

[8] B. I. Henry, T. A. M. Langlands, Peter Straka Phys. Rev. Lett. 170602 2010

[9] T.A.M. Langlands, B.I. Henry J. Comput. Phys. 719 736 2005

[10] Vicenc Mendez, Sergei Fedotov, Werner Horsthemke. Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities. Springer Berlin/Heidelberg, 1st edition, 2010. jun

[11] Christopher N Angstmann, I C Donnelly, B.I. Henry Math. Model. Nat. Phenom. 17 27 2013

[12] Peter Straka, Sergei Fedotov J. Theor. Biol. 71 83 2015

[13] Sergei Fedotov, Nickolay Korabel Phys. Rev. E 062127 2015

[14] Aleksander Stanislavsky, Karina Weron, Aleksander Weron Diffusion and relaxation controlled by tempered α-stable processes Phys. Rev. E 6 11 2008

[15] Rina Schumer, David A Benson, Mark M. Meerschaert, Boris Baeumer Water Resour. Res. 2003

[16] Peter Straka, B I Henry Stoch. Process. their Appl. 324 336 2011

[17] A. Jurlewicz, P. Kern, Mark M. Meerschaert, H.P. P. Scheffler Comput. Math. with Appl. 3021 3036 2012

[18] Marcin Magdziarz, H.P. Scheffler, Peter Straka, P. Zebrowski Stoch. Process. their Appl. 4021 4038 2015

[19] K. Weron, A. Jurlewicz, Marcin Magdziarz, A. Weron, J. Trzmiel Phys. Rev. E 1 7 2010

[20] Mark M. Meerschaert, Peter Straka Ann. Probab. 1699 1723 2014

[21] Ofer Busani. Finite Dimensional Fokker-Planck Equations for Continuous Time Random Walks. arXiv 1510.01150, oct 2015.

[22] Mark M. Meerschaert, Peter Straka J. Stat. Phys. 878 886 2012

[23] A. Baule, R. Friedrich Europhys. Lett. 10002 2007

[24] Mark M. Meerschaert, Alla Sikorskii. Stochastic models for fractional calculus. De Gruyter, Berlin/Boston, 2011.

[25] D. Applebaum. Lévy Processes and Stochastic Calculus, volume 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2nd edition, may 2009.

[26] Jean Bertoin. Subordinators: examples and applications, volume 1717 of Lecture Notes in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 1999.

[27] M Mark Meerschaert, Peter Straka. Inverse Stable Subordinators Math. Model. Nat. Phenom. 1 16 2013

[28] A. Weronn Marcin Magdziarz Phys. Rev. E 1 6 2008

[29] Christopher N Angstmann, I C Donnelly, B.I. Henry, T.A.M. Langlands, Peter Straka SIAM J. Appl. Math. 1445 1468 2015

[30] Boris Baeumer, Peter Straka. Fokker–Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits. Proc. Amer. Math. Soc., arXiv 1501.00533, jan 2016.

[31] Marcin Magdziarz, Janusz Gajda, Tomasz Zorawik J. Stat. Phys. 1241 1250 2014

[32] Joseph Horowitz Z. Wahrsch. Verw. Geb. 167 193 1972

[33] Jan Rosiński Stoch. Process. Appl. 677 707 2007

[34] Peter Straka. Continuous Time Random Walk Limit Processes: Stochastic Models for Anomalous Diffusion, available at http://unsworks.unsw.edu.au/fapi/datastream/unsworks:9800/SOURCE02. PhD thesis, University of New South Wales, 2011.

[35] Janusz Gajda, Marcin Magdziarz Phys. Rev. E 1 6 2010

[36] Eli Barkai, Yuan-Chung Cheng J. Chem. Phys. 6167 2003

[37] Ofer Busani. Aging uncoupled continuous time random walk limits. arXiv, 1402.3965, feb 2015.

[38] Christopher N Angstmann, Isaac C Donnelly, Bruce I Henry, Ba Jacobs, Trevor Am Langlands, James A Nichols J. Comput. Phys. 508 534 2016

[39] Jean Jacod and Albert N Shiryaev. Limit Theorems for Stochastic Processes. Springer, dec 2002.

[40] P. Billingsley. Convergence of Probability Measures. Wiley Series in Probability and Statistics. John Wiley Sons Inc, New York, second edition, jan 1968.

[41] Edwin Hewitt Am. Math. Mon. 419 1960

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