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.


82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
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[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), Imperial College Press-World Scientific London, [Chapter 12], pp. 199-229
[3] Barkai, E., Fractional fokker – planck equation, solution, and application, Phys rev E, 63, (2001), 046118-1/18
[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] ()
[7] Bochner, S., Harmonic analysis and the theory of probability, (1955), University of California Press Berkeley · 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), Methuen London · Zbl 0168.16106
[10] Feller, W., An introduction to probability theory and its applications, vol. 2, (1971), Wiley New York · Zbl 0219.60003
[11] Gel‘fand, I.M.; Shilov, G.E., Generalized functions, vol. I, (1964), Academic Press New York and London
[12] Gorenflo, R.; Abdel-Rehim, E., From power laws to fractional diffusion: the direct way, Viet J math, 32, SI, 65-75, (2004) · Zbl 1086.60049
[13] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-276, Reprinted in
[14] Gorenflo, R.; Mainardi, F., Fractional diffusion processes: probability distributions and continuous time random walk, (), 148-166
[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, (), 171-180 · 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] ()
[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), H. Steinhaus Center for Stochastic Methods in Science and Technology Technical University Wroclaw, Poland
[24] Janicki, A.; Weron, A., Simulation and chaotic behavior of α-stable stochastic processes, (1994), Marcel Dekker New York
[25] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · 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), Reprinted in: · 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)
[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), 041103-1/4
[33] Meerschaert, M.M.; Benson, D.A.; P Scheffler, H.; Becker-Kern, P., Governing equations and solutions of anomalous random walk limits, Phys rev E, 66, (2002), 060102-1/4
[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, (), 1-121
[39] Montroll, E.W.; Weiss, G.H., Random walks on lattices II, J math phys, 6, 167-181, (1965) · Zbl 1342.60067
[40] Montroll, E.W.; West, B.J., On an enriched collection of stochastic processes, (), 61-175
[41] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[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), Gordon and Breach New York · Zbl 0818.26003
[45] Sato, K-I., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press Cambridge · Zbl 0973.60001
[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), 011107-1/8
[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), 011104-1/10
[51] Sokolov, I.M., Thermodynamics and fractional fokker – planck equation, Phys. rev. E, 63, (2001), 056111-1/8
[52] Sokolov, I.M., Solutions of a class of non-Markovian fokker – planck equations, Phys rev E, 66, (2002), 041101-1/5
[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), VSP Utrecht · Zbl 0944.60006
[60] Weiss, G.H., Aspects and applications of random walks, (1994), North-Holland Amsterdam
[61] West, B.J.; Bologna, M.; Grigolini, P., Physics of fractal operators, (2003), Springer Verlag New York
[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), Oxford University Press Oxford · Zbl 1080.37082
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