Fractional Dynamics at Multiple Times

Journal of Statistical Physics

Volumn 149, Issue 5, 2012, Pages 878-886

  Meerschaert, Mark M ^a   Straka, Peter ^b  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Anomalous diffusion; Continuous time random walk; Fractional derivative; Fractional dynamics; Levy process; Renewal theory; Subdiffusion; Subordination; Time change

Indexed keywords

EID: 84870732060     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-012-0638-z     Document Type: Article

References (30)

1. Meerschaert, M. M., Sikorskii, A.: Stochastic Models for Fractional Calculus. de Gruyter, Berlin (2012).
2. Montroll, E. W., Weiss, G. H.: Random walks on lattices. II. J. Math. Phys. 6(2), 167-181 (1965).
3. Scher, H., Montroll, E. W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 2, 2455-2477 (1975).
4. Metzler, R., Klafter, J.: The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1-77 (2000).
5. Benson, D. A., Meerschaert, M. M.: A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv. Water Resour. 32(4), 532-539 (2009).
6. Scalas, E.: Five years of continuous-time random walks in econophysics. In: The Complex Networks of Economic Interactions, vol. 567, pp. 3-16 (2006).
7. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1968).
8. Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd edn. Wiley, New York (1971).
9. Skorohod, A. V.: Limit theorems for stochastic processes with independent increments. Teor. Veroâtn. Ee Primen. 2, 145-177 (1957).
10. Meerschaert, M. M., Scheffler, H.-P.: Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 638(3), 623-638 (2004).
11. Straka, P., Henry, B. I.: Lagging and leading coupled continuous time random walks, renewal times and their joint limits. Stoch. Process. Appl. 121(2), 324-336 (2011).
12. Saichev, A. I., Zaslavsky, G. M.: Fractional kinetic equations: solutions and applications. Chaos 7(4), 753-764 (1997).
13. Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61(1), 132 (2000).
14. Baeumer, B., Meerschaert, M. M.: Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4(4), 481-500 (2001).
15. Henry, B. I., Langlands, T. A. M., Straka, P.: Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces. Phys. Rev. Lett. 105(17), 170602 (2010).
16. Baule, A., Friedrich, R.: A fractional diffusion equation for two-point probability distributions of a continuous-time random walk. Europhys. Lett. 77, 10002 (2007).
17. Meerschaert, M. M., Straka, P.: Semi-Markov approach to continuous time random walk limit processes. arXiv: 1206. 1960 (2012).
18. Bertoin, J.: Subordinators: examples and applications In: Lect. Probab. Theory Stat., pp. 1-91 (2004).
19. Barkai, E., Cheng, Y. C.: Aging continuous time random walks. J. Chem. Phys. 118(14), 6167 (2003).
20. Meerschaert, M. M., Scheffler, H.-P.: Triangular array limits for continuous time random walks. Stoch. Process. Appl. 118(9), 1606-1633 (2008).
21. Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117(6), 677-707 (2007).
22. Meerschaert, M. M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusions in heterogeneous systems. Geophys. Res. Lett. 35, L17403-L17407 (2008).
23. Stanislavsky, A., Weron, K., Weron, A.: Diffusion and relaxation controlled by tempered α-stable processes. Phys. Rev. E 78(5), 6-11 (2008).
24. Chechkin, A. V., Hofmann, M., Sokolov, I. M.: Continuous-time random walk with correlated waiting times. Phys. Rev. E 80(3), 031112 (2009).
25. Magdziarz, M., Metzler, R., Szczotka, W., Zebrowski, P.: Correlated continuous-time random walks-scaling limits and Langevin picture. J. Stat. Mech. 2012, P04010 (2012).
26. Tejedor, V., Metzler, R.: Anomalous diffusion in correlated continuous time random walks. J. Phys. A, Math. Theor. 43, 082002 (2010).
27. Barkai, E., Sokolov, I. M.: Multi-point distribution function for the continuous time random walk. J. Stat. Mech. 2007(08), P08001 (2007).
28. Zaburdaev, V. Y.: Microscopic approach to random walks. J. Stat. Phys. 133(1), 159-167 (2008).
29. Politi, M., Kaizoji, T., Scalas, E.: Full characterization of the fractional Poisson process. Europhys. Lett. 96(2), 20004 (2011).
30. Jurlewicz, A., Kern, P., Meerschaert, M. M., Scheffler, H.-P.: Fractional governing equations for coupled random walks. Comput. Math. Appl. 64(10), 3021-3036 (2012).