Stochastic model for ultraslow diffusion

Stochastic Processes and their Applications

Volumn 116, Issue 9, 2006, Pages 1215-1235

  Meerschaert, Mark M ^a   Scheffler, Hans Peter ^b  

^a UNIVERSITY OF OTAGO   (New Zealand)

^b UNIVERSITY OF SIEGEN   (Germany)

Author keywords

Anomalous diffusion; Continuous time random walk; Slowly varying tails; Stable subordinator

Indexed keywords

DIFFUSION; FUNCTION EVALUATION; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILITY DENSITY FUNCTION;

DIFFUSION EQUATION; RANDOM WALKS; STOCHASTIC FOUNDATIONS; STOCHASTIC MODELS;

RANDOM PROCESSES;

EID: 33746632567     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2006.01.006     Document Type: Article

References (51)

1. Arendt W., Batty C., Hieber M., and Neubrander F. Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics (2001), Birkhaeuser-Verlag, Berlin
2. Baeumer B., and Meerschaert M. Stochastic solutions for fractional Cauchy problems. Frac. Calc. Appl. Anal. 4 (2001) 481-500
3. Baeumer B., Benson D.A., and Meerschaert M. Advection and dispersion in time and space. Phys. A 350 (2005) 245-262
4. Becker-Kern P., Meerschaert M.M., and Scheffler H.P. Limit theorem for continuous time random walks with two time scales. J. Appl. Probab. 41 2 (2004) 455-466
5. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. The fractional-order governing equation of Lévy motion. Water Resour. Res. 36 (2000) 1413-1424
6. Berg C., Reus Christensen J.P., and Ressel P. Harmonic Analysis on Semigroups (1984), Springer-Verlag, New York
7. Bingham N.H. Limit theorems for occupation times of Markov processes. Z. Wharsch. verw. Geb. 17 (1971) 1-22
8. Bisquert J. Fractional diffusion in the multiple-trapping regime and revision of the equivalence with the continuous-time random walk. Phys. Rev. Lett. 91 1 (2003) 602-605
9. Caputo M. Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 13 (1967) 529-539
10. Chaves A. A fractional diffusion equation to describe Lévy flights. Phys. Lett. A 239 (1998) 13-16
11. Chechkin A.V., Klafter J., and Sokolov I.M. Fractional Fokker-Plank equation for ultraslow kinetics. Europhys. Lett. 63 3 (2003) 326-332
12. Chechkin A.V., Gorenflo R., Sokolov I.M., and Gonchar V.Yu. Distributed order time fractional diffusion equation. Frac. Calc. Appl. Anal. 6 3 (2003) 259-279
13. Chechkin A.V., Gorenflo R., and Sokolov I.M. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66 (2002) 046129-046135
14. Darling D.A. The influence of the maximum term in the addition of independent random variable. Trans. Amer. Math. Soc. 73 (1952) 95-107
15. Dräger J., and Klafter J. Strong anomaly in diffusion generated by iterated maps. Phys. Rev. Lett. 84 (2000) 5998-6001
16. Dwass M. Extremal processes. Ann. Math. Statist. 35 4 (1964) 1718-1725
17. Feller W. An Introduction to Probability Theory and Its Applications. 2nd ed. vol. II (1971), Wiley, New York
18. Fontes L.R.G., Isope M., and Newman C.M. Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization, and aging in one dimension. Ann. Probab. 30 2 (2002) 579-604
19. Gorenflo R., and Mainardi F. Fractional diffusion processes: Probability distribution and continuous time random walk. Lecture Notes in Phys. 621 (2003) 148-166
20. Iglói F. Anomalous diffusion in aperiodic environments. Phys. Rev. E 59 (1999) 1465-1474
21. Jacob N. Pseudo-Differential Operators and Markov Processes (1996), Akad. Verl., Berlin
22. Jurek Z., and Mason J.D. Operator-Limit Distributions in Probability Theory (1993), Wiley, New York
23. Kasahara Y. A limit theorem for sums of i.i.d. random variables with slowly varying tail probability. J. Math. Kyoto Univ. 26 3 (1986) 437-443
24. Klafter J., Blumen A., and Shlesinger M.F. Stochastic pathways to anomalous diffusion. Phys. Rev. A 35 (1987) 3081-3085
25. Kozubowski T.J., Meerschaert M.M., and Scheffler H.P. ν-operator stable laws. Publ. Math. Debrecen. 63 4 (2003) 569-585
26. Meerschaert M.M., Benson D.A., and Baeumer B. Multidimensional advection and fractional dispersion. Phys. Rev. E 59 (1999) 5026-5028
27. Meerschaert M.M., Benson D.A., and Baeumer B. Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63 (2001) 1112-1117
28. Meerschaert M.M., and Scheffler H.P. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice (2001), Wiley Interscience, New York
29. Meerschaert M.M., Benson D.A., Scheffler H.P., and Baeumer B. Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65 (2002) 1103-1106
30. Meerschaert M.M., and Scheffler H.P. Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41 3 (2004) 623-638
31. Meerschaert M.M., and Scheffler H.P. Limit theorems for continuous time random walks with slowly varying waiting times. Statist. Probab. Lett. 71 1 (2005) 15-22
32. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
33. Montroll E.W., and Weiss G.H. Random walks on lattices. II. J. Mathematical Phys. 6 (1965) 167-181
34. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential equations. Applied Mathematical Sciences vol. 44 (1983), Springer-Verlag, New York
35. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
36. Prosen T., and Znidaric M. Anomalous diffusion and dynamical localization in polygonal billiards. Phys. Rev. Lett. 87 (2001) 114101-114104
37. Ressel P. Semigroups in probability theory. In: Heyer H. (Ed). Probability Measures on Groups. X. Proceedings of the Tenth Conference held in Oberwolfach, November 4-10 (1990), Plenum Press, New York
38. Saichev A.I., and Zaslavsky G.M. Fractional kinetic equations: Solutions and applications. Chaos 7 (1997) 753-764
39. Samorodnitsky G., and Taqqu M. Stable non-Gaussian Random Processes (1994), Chapman and Hall, New York
40. Sato K.I. Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press
41. Samko S., Kilbas A., and Marichev O. Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach, London
42. Scher H., and Lax M. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7 (1973) 4491-4502
43. Schiessel H., Sokolov I.M., and Blumen A. Dynamics of a polyampholyte hooked around an obstacle. Phys. Rev. E 56 (1997) R2390-R2393
44. Shlesinger M., Klafter J., and Wong Y.M. Random walks with infinite spatial and temporal moments. J. Statist. Phys. 27 (1982) 499-512
45. Seneta E. Regularly Varying Functions. Lecture Notes Mathematics vol. 508 (1976), Springer-Verlag, Berlin
46. Sinai Y.G. The limiting behavior of a one-dimensional random walk in a random medium. Theor. Probab. Appl. 27 (1982) 256-268
47. Sokolov I.M., Chechkin A.V., and Klafter J. Fractional diffusion equation for power-law-truncated Lévy processes. Phys. A 336 (2004) 245-251
48. Sokolov I.M., Chechkin A.V., and Klafter J. Distributed-order fractional kinetics. Acta Phys. Polon. B 35 (2004) 1323
49. Uchaikin V.V., and Zolotarev V.M. Chance and Stability. Stable Distributions and Their Applications (1999), VSP, Utrecht
50. Watanabe S. A limit theorem for sums of non-negative i.i.d. random variables with slowly varying tail probabilities. Proc. 5th International Conf. Multivariate Anal. (1980), North-Holland, Amsterdam 249-261
51. Zaslavsky G. Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Phys. D 76 (1994) 110-122