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Continuous-time random walk and parametric subordination in fractional diffusion. (English) Zbl 1142.82363
Summary: The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit, we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Lévy process, we generate and display sample paths for some special cases.

##### MSC:
 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. 82C70 Transport processes (time-dependent statistical mechanics)
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##### References:
 [1] Baeumer, B.; Meerschaert, M. M.: Stochastic solutions for fractional Cauchy problems. Fract calcul appl anal 4, 481-500 (2001) · Zbl 1057.35102 [2] Balescu, R.: Statistical dynamics: matter out of equilibrium. (1994) · Zbl 0997.82505 [3] Barkai, E.: Fractional Fokker -- Planck equation, solution, and application. Phys rev E 63 (2001) [4] Barkai, E.: CTRW pathways to the fractional diffusion equation. Chem phys 284, 13-27 (2002) [5] Barkai, E.; Metzler, R.; Klafter, J.: From continuous time random walk to fractional Fokker -- Planck equation. Phys rev E 61, 132-138 (2000) [6] Barndorff-Nielsen, O. E.; Mikosch, T.; Resnick, S. I.: Lévy processes: theory and applications. (2001) · Zbl 0961.00012 [7] Bochner, S.: Harmonic analysis and the theory of probability. (1955) · Zbl 0068.11702 [8] Bochner, S.: Subordination of non-Gaussian stochastic processes. Proc natl acad sci USA 48, 19-22 (1962) · Zbl 0105.33002 [9] Cox, D. R.: Renewal theory. (1967) · Zbl 0168.16106 [10] Feller, W.: An introduction to probability theory and its applications. 2 (1971) · Zbl 0219.60003 [11] Gel&grave, I. M.; Fand; Shilov, G. E.: Generalized functions. (1964) [12] Gorenflo, R.; Abdel-Rehim, E.: From power laws to fractional diffusion: the direct way. Viet J math 32, No. SI, 65-75 (2004) · Zbl 1086.60049 [13] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. Fractals and fractional calculus in continuum mechanics, 223-276 (1997) [14] Gorenflo, R.; Mainardi, F.: Fractional diffusion processes: probability distributions and continuous time random walk. Lecture notes in physics 621, 148-166 (2003) [15] Gorenflo R, Mainardi F, Simply and multiply scaled diffusion limits for continuous time random walks. In: Benkadda S, Leoncini X, Zaslavsky G. editors. Proceedings of the international workshop on chaotic transport and complexity in fluids and plasmas Carry Le Rouet (France) 20 -- 25 June 2004, IOP (Institute of Physics) Journal of Physics: Conference Series 7; 2005. p. 1 -- 16. [16] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M.: Fractional calculus and continuous-time finance III: The diffusion limit. Mathematical finance, 171-180 (2001) · Zbl 1138.91444 [17] Grigolini, P.; Rocco, A.; West, B. J.: Fractional calculus as a macroscopic manifestation of randomness. Phys rev E 59, 2603-2613 (1999) [18] Hilfer, R.: Exact solutions for a class of fractal time random walks. Fractals 3, 211-216 (1995) · Zbl 0881.60066 [19] Hilfer, R.: Applications of fractional calculus in physics. (2000) · Zbl 0998.26002 [20] Hilfer, R.: On fractional diffusion and continuous time random walks. Physica A 329, 35-39 (2003) · Zbl 1029.60033 [21] Hilfer, R.; Anton, L.: Fractional master equations and fractal time random walks. Phys rev E 51, R848-R851 (1995) [22] Jacob N. Pseudodifferential operators -- Markov processes, vol. I: Fourier analysis and semigroups vol. II: Generators and their potential theory, vol. III: Markov Processes and Applications, Imperial College Press, London (2001), (2002), (2005). [23] Janicki, A.: Numerical and statistical approximation of stochastic differential equations with non-Gaussian measures monograph, no. 1. (1996) [24] Janicki, A.; Weron, A.: Simulation and chaotic behavior of $\alpha$-stable stochastic processes. (1994) · Zbl 0955.60508 [25] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations. (2006) · Zbl 1092.45003 [26] Kotulski, M.: Asymptotic distributions of continuous-time random walks: a probabilistic approach. J stat phys 81, 777-792 (1995) · Zbl 1107.60318 [27] Mainardi, F.; Luchko, Yu.; Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract calcul appl anal 4, 153-192 (2001) · Zbl 1054.35156 [28] Mainardi, F.; Pagnini, G.; Gorenflo, R.: Mellin transform and subordination laws in fractional diffusion processes. Fract calcul appl anal 6, 441-459 (2003) · Zbl 1083.60032 [29] Mainardi, F.; Pagnini, G.; Saxena, R. K.: Fox H functions in fractional diffusion. J computat appl math 178, 321-331 (2005) · Zbl 1061.33012 [30] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E.: Fractional calculus and continuous-time finance II: The waiting-time distribution. Physica A 287, 468-481 (2000) · Zbl 1138.91444 [31] Mannella, R.; Grigolini, P.; West, B. J.: A dynamical approach to fractional Brownian motion. Fractals 2, 81-94 (1994) [32] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. P.; Baeumer, B.: Stochastic solutions of space-fractional diffusion equation. Phys rev E 65 (2002) · Zbl 1244.60080 [33] Meerschaert, M. M.; Benson, D. A.; Scheffler, H. P.; Becker-Kern, P.: Governing equations and solutions of anomalous random walk limits. Phys rev E 66 (2002) [34] Metzler, R.; Klafter, J.; Sokolov, I. M.: Anomalous transport in external fields: continuous time random walks and fractional diffusion equations extended. Phys rev E 58, 1621-1633 (1998) [35] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys rep 339, 1-77 (2000) · Zbl 0984.82032 [36] Metzler, R.; 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, R161-R208 (2004) · Zbl 1075.82018 [37] Montroll, E. W.; Scher, H.: Random walks on lattices IV: Continuous-time walks and influence of absorbing boundaries. J stat phys 9, 101-135 (1973) [38] Montroll, E. W.; Shlesinger, M. F.: On the wonderful world of random walks. Nonequilibrium phenomena II: From stochastics to hydrodynamics, 1-121 (1984) [39] Montroll, E. W.; Weiss, G. H.: Random walks on lattices II. J math phys 6, 167-181 (1965) [40] Montroll, E. W.; West, B. J.: On an enriched collection of stochastic processes. Fluctuation phenomena, 61-175 (1979) [41] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008 [42] Piryatinska, A.; Saichev, A. I.; Woyczynski, W. A.: Models of anomalous diffusion: the subdiffusive case. Physica A 349, 375-420 (2005) [43] Saichev, A.; Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos 7, 753-764 (1997) · Zbl 0933.37029 [44] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003 [45] Sato, K-I.: Lévy processes and infinitely divisible distributions. (1999) [46] Scalas, E.: The application of continuous-time random walks in finance and economics. Physica A 362, 225-239 (2006) [47] Scalas, E.; Gorenflo, R.; Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376-384 (2000) [48] Scalas, E.; Gorenflo, R.; Mainardi, F.: Uncoupled continuous-time random walks: solution and limiting behavior of the master equation. Phys rev E 69 (2004) · Zbl 1087.60064 [49] Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J.: Strange kinetics. Nature 363, 31-37 (1993) [50] Sokolov, I. M.: Lévy flights from a continuous-time process. Phys rev E 63 (2001) [51] Sokolov, I. M.: Thermodynamics and fractional Fokker -- Planck equation. Phys. rev. E 63 (2001) [52] Sokolov, I. M.: Solutions of a class of non-Markovian Fokker -- Planck equations. Phys rev E 66 (2002) [53] Sokolov, I. M.; Klafter, J.; Blumen, A.: Linear response in complex systems: CTRW and the fractional Fokker -- Planck equations. Physica A 302, 268-278 (2001) · Zbl 0983.60040 [54] Sokolov, I. M.; Klafter, J.: From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos 15, 026103-026109 (2005) · Zbl 1080.82022 [55] Sokolov, I. M.; Klafter, J.; Blumen, A.: Fractional kinetics. Phys today 55, 48-54 (2002) [56] Stanislavski, A. A.: Memory effects and macroscopic manifestation of randomness. Phys rev E 61, 4752-4759 (2000) [57] Stanislavsky, A. A.: Black -- Scholes model under subordination. Physica A 318, 469-474 (2003) · Zbl 1010.91029 [58] Uchaikin, V. V.; Saenko, V. V.: Stochastic solution of partial differential equations of fractional orders. Siber J numer math 6, 197-203 (2003) · Zbl 1032.60057 [59] Uchaikin, V. V.; Zolotarev, V. M.: Chance and stability stable distributions and their applications. (1999) · Zbl 0944.60006 [60] Weiss, G. H.: Aspects and applications of random walks. (1994) · Zbl 0925.60079 [61] West, B. J.; Bologna, M.; Grigolini, P.: Physics of fractal operators. (2003) [62] Wyss, M. M.; Wyss, W.: Evolution, its fractional extension and generalization. Fract calcul appl anal 4, 273-284 (2001) · Zbl 1042.45005 [63] Zaslavsky, G. M.: Chaos, fractional kinetics and anomalous transport. Phys rep 371, 461-580 (2002) · Zbl 0999.82053 [64] Zaslavsky, G. M.: Hamiltonian chaos and fractional dynamics. (2005) · Zbl 1083.37002