Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics

Theory of Probability and its Applications

Volumn 53, Issue 4, 2009, Pages 594-609

  Kolokoltsov, V N ^a,b  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b Moscow Economics Institute   (Russian Federation)

Author keywords

Anomalous diffusion; Continuous time random walks; Fractional derivatives; Fractional stable distributions; Hitting time processes; L vy subordinators; Limit theorems; Time change

Indexed keywords

EID: 73549104035     PISSN: 0040585X     EISSN: 10957219     Source Type: Journal    
DOI: 10.1137/S0040585X97983857     Document Type: Article

References (25)

1. R. F. BASS, Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields, 79 (1988), pp. 271-287.
2. P. BECKER-KERN, M. M. MEERSCHAERT, AND H.-P. SCHEFFLER, Limit theorems for coupled continuous time random walks, Ann. Probab., 32 (2004), pp. 730-756.
3. V. BENING, V. KOROLEV, T. SUKHORUKOVA, G. GUSAROV, V. SAENKO, AND V. KOLOKOLTSOV, Fractionally stable distributions, in Stochastic Models of Plasma Turbulence, V. Korolev and N. Skvortsova, eds., VSP, Leiden, 2006, pp. 175-244.
4. V. E. BENING, V. YU. KOROLEV, V. N. KOLOKOLTSOV, V. V. SAENKO, V. V. UCHAIKIN, AND V. M. ZOLOTAREV, Estimation of parameters of fractional stable distributions, J. Math. Sci. (N. Y.), 123 (2004), pp. 3722-3732.
5. V. E. BENING, V. YU. KOROLEV, AND V. N. KOLOKOLTSOV, Limit theorems for continuoustime random walks in the double-array limit scheme, J. Math. Sci. (N. Y.), 138 (2006), pp. 5348-5365.
6. O. KALLENBERG, Foundations of Modern Probability, 2nd ed., Springer-Verlag, Berlin, 2002.
7. M. KOTULSKI, Asymptotic distribution of continuous-time random walks: A probabilistic approach, J. Statist. Phys., 81 (1995), pp. 777-792.
8. V. N. KOLOKOLTSOV, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc., 80 (2000), pp. 725-768.
9. V. N. KOLOKOLTSOV, Semiclassical Analysis for Diffusions and Stochastic Processes, Lecture Notes in Math. 1724, Springer-Verlag, Berlin, 2000.
10. V. N. KOLOKOLTSOV, Nonlinear Markov semigroups and interacting Lév y ty pe processes, J. Stat. Phys., 126 (2007), pp. 585-642.
11. V. N. KOLOKOLTSOV, V. YU. KOROLEV, AND V. V. UCHAIKIN, Fractional stable distributions, J. Math. Sci. (N. Y.), 105 (2001), pp. 2569-2576.
12. V. YU. KOROLEV, V. E. BENING, AND S. YA. SHORGIN, Mathematical Basis of Risk Theory, Fismatlit, Moscow, 2007 (in Russian).
13. V. YU. KOROLEV ET AL., Some methods of the analysis of time characteristics of catastrophes in non-homogeneous flows of extremal events, in Systems and Methods of Informatics, I. Sokolov, ed., Moscow, RAN, 2006, pp. 5-23 (in Russian).
14. V. P. MASLOV, Perturbation Theory and Asymptotical Methods, Moscow State University Press, Moscow, 1965 (in Russian).
15. M. M. MEERSCHAERT AND H.-P. SCHEFFLER, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab., 41 (2004), pp. 623-638.
16. M. M. MEERSCHAERT AND H.-P. SCHEFFLER, Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice, John Wiley and Sons, New York, 2001.
17. R. METZLER AND J. KLAFTER, The random walk's guide to anomalous diffusion: A fractional dynamic approach, Phys. Rep., 339 (2000), pp. 1-77.
18. K. S. MILLER AND B. ROSS, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
19. E. W. MONTROLL AND G. H. WEISS, Random walks on lattices, II, J. Math. Phys., 6 (1965), pp. 167-181.
20. A. I. SAICHEV AND W. A. WOYCZYŃSKI, Distributions in the Physical and Engineering Sciences, Vo l. 1, Birkhäuser Boston, Boston, MA, 1997.
21. A. I. SAICHEV AND G. M. ZASLAVSKY, Fractional kinetic equations: Solutions and applications, Chaos, 7 (1977), pp. 753-764.
22. A. A. SAMARSKII, Theory of Difference Schemes, 3rd ed., Nauka, Moscow, 1989 (in Russian).
23. V. V. UCHAIKIN, Montroll-Weiss problem, fractional equations, and stable distributions, Internat. J. Theoret. Phys., 39 (2000) pp. 2087-2105.
24. W. WHITT, Stochastic-Process Limits, Springer-Verlag, New York, 2002.
25. G. M. ZASLAVSKY, Fractional kinetic equation for Hamiltonian chaos, Phys. D, 76 (1994), pp. 110-122.