Discrete and continuous random walk models for space-time fractional diffusion

Journal of Mathematical Sciences

Volumn 132, Issue 5, 2006, Pages 614-628

  Gorenflo, R ^a   Vivoli, A ^b   Mainardi, F ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 30644458726     PISSN: 10723374     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10958-006-0006-0     Document Type: Conference Paper

References (40)

1. B. Baeumer and M. M. Meerschaert, "Stochastic solutions for fractional Cauchy problems," Frac. Cal. Appl. Anal., 4, 481-500 (2001).
2. M. Caputo, "Linear models of dissipation whose Q is almost frequency independent, Part II," Geophys. J. R. Astr. Soc., 13, 529-539 (1967).
3. M. Caputo, Elasticità e Dissipazione [in Italian], Zanichelli, Bologna (1969).
4. M. Caputo, "Models of flux in porous media with memory," Water Resour. Res., 36, 693-705 (2000).
5. M. Caputo, "Distributed order differential equations modelling dielectric induction and diffusion," Frac. Cal. Appl. Anal., 4, 421-422 (2001).
6. M. Caputo and F. Mainardi, "Linear models of dissipation in anelastic solids," Riv. Nuovo Cimento, 1, 161-198 (1971).
7. D. R. Cox, Renewal Theory, Methuen, London (1967).
8. M. M. Djrbashian and A. B. Nersesian, "Fractional derivatives and the Cauchy problem for differential equations of fractional order," Izv. Acad. Nauk Armjanskvy SSR, Mat., 3, 3-29 (1968).
9. W. Feller, "On a generalization of Marcel Riesz' potentials and the semi-groups generated by them," in: Meddelanden Lunds Universitets Matematiska Seminarium, Lund (1952), pp. 73-81.
10. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York, (1971).
11. J. E. Gillis and G. H. Weiss, "Expected number of distinct sites visited by a random walk with an infinite variance," J. Math. Phys., 11, 1307-1312 (1970).
12. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass. (1954).
13. R. Gorenflo and E. Abdel-Rehim, "From power laws to fractional diffusion," Vietnam J. Math, (to appear).
14. R. Gorenflo, G. De Fabritiis, and F. Mainardi, "Discrete random walk models for symmetric Lévy-Feller diffusion processes," Phys. A, 269, 79-89 (1999).
15. R. Gorenflo, A. Iskenderov, and Yu. Luchko, "Mapping between solutions of fractional diffusion-wave equations," Frac. Calcul Appl. Anal., 3, 75-86 (2000).
16. R. Gorenflo and F. Mainardi, "Fractional calculus: Integral and differential equations of fractional order," in: Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien (1997), pp. 223-276.
17. R. Gorenflo and F. Mainardi, "Random walk models for space-fractional diffusion processes," Frac. Calcul. Appl. Anal., 1, 167-191 (1998).
18. R. Gorenflo and F. Mainardi, "Approximation of Lévy-Feller diffusion by random walk," J. Anal. Appl., 18, 231-246 (1999).
19. R. Gorenflo and F. Mainardi, "Random walk models approximating symmetric space-fractional diffusion processes," in: Problems in Mathematical Physics, Birkhäuser Verlag, Basel (2001), pp. 120-145.
20. R. Gorenflo and F. Mainardi, "Nonmarkovian random walks, scaling and diffusion limits," in: The Second Conference on Lévy Processes: Theory and Applications, Dept. Mathematics, Univ. of Aarhus, Denmark (2002), pp. 120-128.
21. R. Gorenflo and F. Mainardi, "Fractional diffusion processes: Probability distributions and continuous time random walk," in: Processes with Long Range Correlations, Springer, Berlin (2003), pp. 148-166.
22. R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, and P. Paradisi, "Discrete random walk models for space-time fractional diffusion," Chemical Phys., 284, 521-544 (2002).
23. R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, "Time-fractional diffusion: a discrete random walk approach," Nonlinear Dynamics, 29, 129-143 (2002).
24. R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto, "Fractional calculus and continuous-time finance. III: The diffusion limit," in: Mathematical Finance, Birkhäuser Verlag, Basel (2001), pp. 171-180.
25. R. Gorenflo and A. Vivoli, "Fully discrete random walks for space-time fractional diffusion equations," Signal Proc., 83, 2411-2420 (2003).
26. R. Hilfer, "Exact solutions for a class of fractal time random walks," Fractals, 3, 211-216 (1995).
27. R. Hilfer, "On fractional diffusion and continuous time random walks," Physica A, 329, 35-40 (2003).
28. R. Hilfer and L. Anton, "Fractional integral equations and fractal time random walks," Phys. Rev., E 51, 848-851 (1995).
29. A. N. Kochubei, "A Cauchy problem for evolution equations of fractional order," J. Diff. Eqns., 25, 967-974 (1989).
30. A. N. Kochubei, "Fractional order diffusion," J. Diff. Eqns., 26, 485-492 (1990).
31. M. Kotulski, "Asymptotic distributions of continuous-time random walks: a probabilistic approach," J. Statist. Phys., 81, 777-792 (1995).
32. F. Mainardi, Yu. Luchko, and G. Pagnini, "The fundamental solution of the space-time fractional diffusion equation," Frac. Calcul. Appl. Anal., 4, 153-192 (2001).
33. F. Mainardi, M. Raberto, R. Gorenflo, and E. Scalas, "Fractional calculus and continuous-time finance. II: The waiting-time distribution," Physica A, 287, 468-481 (2000).
34. E. W. Montroll and G. H. Weiss, "Random walks on lattices, II, " J. Math. Phys., 6, 167-181 (1965).
35. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999).
36. A. I. Saichev and G. M. Zaslavskii, "Fractional kinetic equations: Solutions and applications," Chaos, 7, 753-764 (1997).
37. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (1993).
38. E. Scalas, R. Gorenflo, and F. Mainardi, "Fractional calculus and continuous-time finance," Physica A, 284, 376-384 (2000).
39. V. V. Uchaikin and V. M. Zolotarev, Chance and Stability: Stable Distributions and Their Applications, VSP, Utrecht (1999).
40. G. H. Weiss, Aspects and Applications of Random Walks, North-Holland, Amsterdam (1994).