Langevin picture of subdiffusion with infinitely divisible waiting times. (English) Zbl 1177.82101
In many instances physics-motivated random transport processes deviate from the customary diffusion scenario. The present paper addresses a subdiffusive transport in the presence of time-dependent forces. The fractional integro-differential Fokker-Planck equation, derived and solved previously [Phys. Rev. Lett. 97, 140602 (2001)] by means of limit distribution of a continuous time random walk, is known to refer to a non-Markov stochastic process. Presently, so-called Langevin picture underlying the subdifffusive F-P dynamics has been proposed. An explicit construction of the pertinent subordinated Langevin process was given. A strongly and uniformly covergent approximation scheme for this process entails an efficient numerical simulation of its sample paths.
Reviewer: Piotr Garbaczewski (Opole)
MSC:
| 82C70 | Transport processes in time-dependent statistical mechanics |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60F05 | Central limit and other weak theorems |
| 60K37 | Processes in random environments |
| 60G52 | Stable stochastic processes |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 26A33 | Fractional derivatives and integrals |
| 46F12 | Integral transforms in distribution spaces |
Keywords:
anomalous diffusion; subdiffusion; continuous time random walks; inverse subordinator; first-passage time; fractional Fokker-Planck equation; infinitely divisible distribution; non-Markovian dynamics; Langevin representationReferences:
| [1] | Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61, 132–138 (2000) · doi:10.1103/PhysRevE.61.132 |
| [2] | Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) · Zbl 0861.60003 |
| [3] | Bertoin, J., van Harn, K., Steutel, F.W.: Renewal theory and level passage by subordinators. Stat. Probab. Lett. 45, 65–69 (1999) · Zbl 0965.60053 · doi:10.1016/S0167-7152(99)00043-7 |
| [4] | Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes, vol. 1, 2nd edn. Springer, New York (2003) · Zbl 1026.60061 |
| [5] | Eliazar, I., Klafter, J.: Lévy-driven Langevin systems: Targeted stochasticity. J. Stat. Phys. 111, 739–768 (2003) · Zbl 1049.82053 · doi:10.1023/A:1022894030773 |
| [6] | Grandell, J.: Doubly Stochastic Poisson Processes. Lecture Notes Math., vol. 529. Springer, Berlin (1976) · Zbl 0339.60053 |
| [7] | Heinsalu, E., Patriarca, M., Goychuk, I., Hänggi, P.: Use and abuse of a fractional Fokker-Planck dynamics for time-dependent driving. Phys. Rev. Lett. 99, 120602 (2007) · doi:10.1103/PhysRevLett.99.120602 |
| [8] | Janicki, A., Weron, A.: Simulation and Chaotic Behaviour of {\(\alpha\)}-Stable Stochastic Processes. Marcel Dekker, New York (1994) · Zbl 0955.60508 |
| [9] | Jurlewicz, A., Weron, K., Teuerle, M.: Generalized Mittag-Leffler relaxation: Clustering-jump continuous-time random walk approach. Phys. Rev. E 78, 011103 (2008) · doi:10.1103/PhysRevE.78.011103 |
| [10] | Kotz, S., Kozubowski, T.J., Podgórski, K.: The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering and Finance. Birkhäuser, Boston (2001) · Zbl 0977.62003 |
| [11] | Lagerås, A.N.: A renewal-process-type expression for the moments of inverse subordinators. J. Appl. Probab. 42, 1134–1144 (2005) · Zbl 1094.60057 · doi:10.1239/jap/1134587822 |
| [12] | Magdziarz, M.: Stochastic representation of subdiffusion processes with time-dependent drift. Stoch. Proc. Appl., submitted (2008) · Zbl 1180.60051 |
| [13] | Magdziarz, M., Weron, A., Klafter, J.: Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: The case of a time-dependent force. Phys. Rev. Lett. 101, 210601 (2008) · doi:10.1103/PhysRevLett.101.210601 |
| [14] | Meerschaert, M.M., Scheffler, H.P.: Stochastic model for ultraslow diffusion. Stoch. Proc. Appl. 116, 1215–1235 (2006) · Zbl 1100.60024 · doi:10.1016/j.spa.2006.01.006 |
| [15] | Meerschaert, M.M., Benson, D.A., Scheffler, H.P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 041103 (2002) · Zbl 1244.60080 · doi:10.1103/PhysRevE.65.041103 |
| [16] | 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 · doi:10.1016/S0370-1573(00)00070-3 |
| [17] | Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999) · doi:10.1103/PhysRevLett.82.3563 |
| [18] | Montroll, E.W., Weiss, G.H.: Random walks on lattices II. J. Math. Phys. 6, 167–181 (1965) · Zbl 1342.60067 · doi:10.1063/1.1704269 |
| [19] | Rosinski, J.: Simulation of Lévy processes. In: Encyclopedia of Statistics in Quality and Reliability: Computationally Intensive Methods and Simulation. Wiley, New York (2008) |
| [20] | Samko, S.G., Kilbas, A.A., Maritchev, D.I.: Integrals and Derivatives of the Fractional Order and Some of Their Applications. Gordon and Breach, Amsterdam (1993) |
| [21] | Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) |
| [22] | Sokolov, I.M., Klafter, J.: Field-induced dispersion in subdiffusion. Phys. Rev. Lett. 97, 140602 (2006) · doi:10.1103/PhysRevLett.97.140602 |
| [23] | Stanislavsky, A.A., Weron, K., Weron, A.: Diffusion and relaxation controlled by tempered-stable processes. Phys. Rev. E 78, 051106 (2008) · Zbl 1164.82007 · doi:10.1103/PhysRevE.78.051106 |
| [24] | van Harn, K., Steutel, F.W.: Stationarity of delayed subordinators. Stoch. Models 17, 369–374 (2001) · Zbl 0988.60048 · doi:10.1081/STM-100002278 |
| [25] | Weron, R.: On the Chambers-Mallows-Stuck method for simulating skewed stable random variables. Stat. Probab. Lett. 28, 165–171 (1996) · Zbl 0856.60022 · doi:10.1016/0167-7152(95)00113-1 |
| [26] | Winkel, M.: Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Probab. 42, 138–152 (2005) · Zbl 1078.60035 · doi:10.1239/jap/1110381376 |
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