Non-Markovian diffusion equations and processes: Analysis and simulations

Physica A: Statistical Mechanics and its Applications

Volumn 387, Issue 21, 2008, Pages 5033-5064

  Mura, A ^a   Taqqu, M S ^b   Mainardi, F ^a  

^a UNIVERSITY OF BOLOGNA   (Italy)

^b Boston University   (United States)

Author keywords

Anomalous diffusion; Fractional Brownian motion; Fractional derivatives; Non Markovian processes; Subordination

Indexed keywords

DENSITY FUNCTIONAL THEORY; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; DIFFUSION; FOKKER PLANCK EQUATION; LEARNING SYSTEMS; LINEAR EQUATIONS; MATHEMATICAL MODELS; RANDOM PROCESSES; SEMICONDUCTOR DOPING; STOCHASTIC MODELS; STOCHASTIC PROGRAMMING;

(E ,3E) PROCESS; DENSITY FUNCTIONS; DIFFUSION EQUATIONS (DE); DIFFUSION TYPE EQUATIONS; ELSEVIER (CO); EVOLUTION EQUATIONS; EXACT SOLUTIONS; FOKKER PLANCK EQUATION (FPE); MARKOVIAN; MEMORY KERNELS; NON-MARKOVIAN; PATH SIMULATIONS; TIME EVOLUTIONS;

FUNCTION EVALUATION;

EID: 47649112182     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2008.04.035     Document Type: Article

References (53)

1. Barkai E. CTRW pathways to the fractional diffusion equation. Chem. Phys. 284 (2002) 13-27
2. Barkai E., Metzler R., and Klafter J. From continuous time random walk to fractional Fokker-Planck equation. Phys. Rev. E 61 (2000) 132-138
3. Beghin L., and Orsingher E. The distribution of the local time for pseudoprocesses and its connection with fractional diffusion equations. Stochastic Process. Appl. 115 (2005) 1017-1040
4. Bochner S. Harmonic Analysis and the Theory of Probability (1955), University of California Press, Berkeley
5. Bochner S. Subordination of non-Gaussian stochastic processes. Proc. Natl. Acad. Sci., USA 48 (1962) 19-22
6. Dentz M., et al. Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport. Adv. Water Resources 27 (2004) 155-173
7. C. Donati-Martin, B. Roynette, P. Vallois, M. Yor, On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension d = 2 (1 - α); 0 < α < 1, Studia Scientiarum Mathematicarum Hungarica, (in press) (2007). E-print http://hal.archives-ouvertes.fr/hal-00141513/en/
8. Feller W. An Introduction to Probability Theory and its Applications, vol. 1. 3-rd ed. (1968), Wiley, New York 1-st ed. 1957
9. Feller W. An Introduction to Probability Theory and its Applications, vol. 2. 2-nd ed. (1971), Wiley, New York 1-st ed. 1966
10. Fox C. The G and H-Functions as Symmetrical Fourier Kernels. Trans. Amer. Math. Soc. 98 (1961) 395-429
11. Gorenflo R., Luchko Yu., and Mainardi F. Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2 (1999) 383-414. E-print http://arxiv.org/abs/math-ph/0701069
12. Gorenflo R., and Mainardi F. Fractional diffusion processes: Probability distributions and continuous time random walk. In: Rangarajan G., and Ding M. (Eds). Processes with Long Range Correlations. Lecture Notes in Physics vol. 621 (2003), Springer Verlag, Berlin 148-161
13. Gorenflo R., Mainardi F., Scalas E., and Raberto M. Fractional calculus and continuous-time finance III: The diffusion limit. In: Kohlmann M., and Tang S. (Eds). Mathematical Finance (2001), Birkhauser, Basel, Switzerland 171-180
14. R. Gorenflo, F. Mainardi, Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: A mathematical approach, in: R. Klages, G. Radons, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim, 2008, pp. 93-127 (E-print) http://arxiv.org/abs/0705.079
15. Gorenflo R., Mainardi F., and Vivoli A. Continuous time random walk and parametric subordination in fractional diffusion. Chaos, Solitons Fractals 34 (2007) 87-103. E-print http://arxiv.org/abs/cond-mat/0701126
16. Grigolini P., Rocco A., and West B.J. Fractional calculus as a macroscopic manifestation of randomness. Phys. Rev. E 59 (1999) 2603-2613
17. Hida T. Brownian Motion (1980), Springer-Verlag, New York, Heidelberg, Berlin
18. Hilfer R., and Anton L. Fractional master equations and fractal time random walks. Phys. Rev. E 51 (1995) R848-R851
19. Kenkre V.M., and Knox R.S. Generalized-master-equation theory of excitation transfer. Phys. Rev. B 9 (1974) 5279-5290
20. D. Kleinhans, R. Friedrich, Continuous Time Random Walks: Simulation of continuous trajectories, E-print http://arxiv.org/abs/0707.3221, pp. 7
21. Kolsrud T. On a class of probabilistic integrodifferential equations. In: Albeverio S., Fenstad J.E., Holden H., and Lindstrøm T. (Eds). Ideas and Methods in Mathematical Analysis, Stochastics and Applications, vol. I (1990), Cambridge University Press, Cambridge 168-172
22. Lee M.L.T., and Whitmore G.A. Stochastic processes directed by randomized time. J. Appl. Probab. 30 (1993) 302-314
23. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons Fractals 7 (1996) 1461-1477
24. Mainardi F. Fractional calculus: Some basic problems in continuum ad statistical mechanics. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Verlag, Wien, New York 291-348. http://www.fracalmo.org Reprinted in
25. Mainardi F., and Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118 (2000) 283-299
26. Mainardi F., Luchko Yu., and Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Frac. Calc. Appl. Anal. 4 (2001) 153-192. E-print http://arxiv.org/abs/cond-mat/0702419
27. Mainardi F., Raberto M., Gorenflo R., and Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A 287 (2000) 468-481
28. Mainardi F., Pagnini G., and Gorenflo R. Mellin transform and subordination laws in fractional diffusion processes. Frac. Calc. Appl. Anal. 6 (2003) 441-459. E-print http://arxiv.org/abs/math/0702133
29. Mainardi F., Pagnini G., and Saxena R.K. Fox H functions in fractional diffusion. J. Comput. Appl. Math. 178 (2005) 321-331
30. Mandelbrot B.B. Note on the definition and stationarity of fractional Gaussian noise. J. Hydrol. 30 (1976) 407-409
31. Mandelbrot B.B., and Van Ness J.W. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422-433
32. Mathai A.M., and Saxena R.K. The H-function with Applications in Statistics and Other Disciplines (1978), Wiley Eastern Ltd, New Delhi
33. M. Meerschaert, B. Baeumer, E. Nane, Brownian subordinators and fractional Cauchy problems, Transactions of the American Mathematical Society (in press)
34. Meerschaert M., Benson D., Scheffler H., and Baeumer B. Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65 (2002) 041103/1-4
35. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
36. Metzler R., and Klafter J. The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. Math. Gen. 37 (2004) R161-R208
37. Molchanov S.A., and Ostrovskii E. Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14 (1969) 128-131
38. A. Mura, F. Mainardi, A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics, (submitted for publication) E-print http://arxiv.org/abs/0711.0665
39. Mura A., and Pagnini G. Characterization and simulation of a class of stochastic processes related to anomalous diffusion. http://arxiv.org/abs/0801.4879 E-print
40. Piryatinska A., Saichev A.I., and Woyczynski W.A. Models of anomalous diffusion: The subdiffusive case. Physica A 349 (2005) 375-420
41. Scalas E., Gorenflo R., and Mainardi F. Fractional calculus and continuous-time finance. Physica A 284 (2000) 376-384
42. Scalas E., Gorenflo R., and Mainardi F. Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E 69 (2004). 011107/1-8. E-print http://arxiv.org/abs/cond-mat/0402657
43. Schneider W.R., and noise G. In: Albeverio S., Fenstad J.E., Holden H., and Lindstrøm T. (Eds). Ideas and Methods in Mathematical Analysis, Stochastics and Applications, vol 1 (1990), Cambridge Univ. Press, Cambridge pp. 261-282
44. Schneider W.R. Fractional diffusion. In: Lima R., Streit L., and Vilela Mendes D. (Eds). Dynamics and Stochastic Processes, Theory and Applications. Lecture Notes in Physics vol. 355 (1990), Springer Verlag, Heidelberg 276-286
45. Sokolov I.M. Solutions of a class of non-Markovian Fokker-Planck equations. Phys. Rev. E 66 (2002) 041101/1-5
46. Srivastava H.M., Gupta K.C., and Goyal S.P. The H-Functions of One and Two Variables with Applications (1982), South Asian Publishers, New Delhi
47. Srokowski T. Non-Markovian Lvy diffusion in non homogeneous media. Phys. Rev. E 75 (2007). 051105/1-8. E-print http://arxiv.org/abs/cond-mat/0611056
48. Stanislavsky A.A. Subordinated brownian motion and its fractional Fokker-Planck equation. Phys. Scr. 67 (2003) 265-268
49. Taqqu M.S. Fractional Brownian motion and long-range dependence. In: Doukhan P., Oppenheim G., and Taqqu M.S. (Eds). Theory and Applications of Long-range Dependence (2003), Birkäuser, Boston, Basel, Berlin 5-38
50. Weiss G.H. Aspects and Applications of Random Walks (1994), North-Holland, Amsterdam
51. Wyss M.M., and Wyss W. Evolution, its fractional extension and generalization. Frac. Calc. Appl. Anal. 4 (2001) 273-284
52. Zaslavsky G.M. Chaos, fractional kinetics and anomalous transport. Phys. Rep. 371 (2002) 461-580
53. Zwanzig R. Memory effects in irreversible thermodynamics. Phys. Rev. 124 (1961) 983-992