Comment on fractional Fokker-Planck equation with space and time dependent drift and diffusion. (English) Zbl 1291.82094
Summary: The purpose of this paper is to correct errors presented recently in [L. Lv et al., J. Stat. Phys. 149, No. 4, 619–628 (2012; Zbl 1260.82064)], where the authors analyzed Fractional Fokker-Planck equation (FFPE) with space-time dependent drift \(F(x,t)=F(x)f(t)\) and diffusion \(D(x,t)=D(x)\tilde d(t)\) coefficients in the factorized form. We show an important drawback in the derivation of the stochastic representation of FFPE presented in the aforementioned paper, which makes the whole proof wrong. Moreover, we present a correct proof of their result in even more general case, when both drift and diffusion can have any, not necessarily factorized, form.
MSC:
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional Fokker-Planck equation; time change; fractional derivative; subdiffusion; stable distributionCitations:
Zbl 1260.82064References:
| [1] | Lv, L., Qiu, W., Ren, F.: Fractional Fokker-Planck equation with space and time dependent drift and diffusion. J. Stat. Phys. 149, 619-628 (2012) · Zbl 1260.82064 · doi:10.1007/s10955-012-0618-3 |
| [2] | Sokolov, I.M., Klafter, J.: Field-induced dispersion in subdiffusion. Phys. Rev. Lett. 97, 140602 (2006) · doi:10.1103/PhysRevLett.97.140602 |
| [3] | Weron, A., Magdziarz, M., Weron, K.: Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation. Phys. Rev. E 77, 036704 (2008) · doi:10.1103/PhysRevE.77.036704 |
| [4] | Magdziarz, M.: Stochastic representation of subdiffusion processes with time-dependent drift. Stoch. Process. Appl. 119, 3238-3252 (2009) · Zbl 1180.60051 · doi:10.1016/j.spa.2009.05.006 |
| [5] | Magdziarz, M.: Langevin picture of subdiffusion with infinitely divisible waiting times. J. Stat. Phys. 135, 763-772 (2009) · Zbl 1177.82101 · doi:10.1007/s10955-009-9751-z |
| [6] | Henry, B.I., Langlands, T.A.M., Straka, P.: Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces. Phys. Rev. Lett. 105, 170602 (2010) · doi:10.1103/PhysRevLett.105.170602 |
| [7] | 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 |
| [8] | 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 |
| [9] | Heinsalu, E., et al.: 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 |
| [10] | Janicki, A., Weron, A.: Simulation and Chaotic Behaviour of \[\alpha\] α-Stable Stochastic Processes. Marcel Dekker, New York (1994) · Zbl 0946.60028 |
| [11] | Sato, K.I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) · Zbl 0973.60001 |
| [12] | Magdziarz, M.: Path properties of subdiffusion-A martingale approach. Stoch. Models 26, 256-271 (2010) · Zbl 1205.60080 |
| [13] | Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004) · Zbl 1073.60002 · doi:10.1017/CBO9780511755323 |
| [14] | Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Vol. 293 of A Series of Comprehensive Studies in Mathematics, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006 |
| [15] | Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
