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Fractional master equation: Non-standard analysis and Liouville-Riemann derivative. (English) Zbl 0994.82062
Summary: Fractional master equations may be defined either by means of Liouville-Riemann fractional derivative or via non-standard analysis. The first approach describes processes with long-range dependence whilst the second approach deals with processes involving independent increments. The present paper puts in evidence some of the differences between these two modellings, and to this end it especially considers more fractional Poisson processes.

MSC:
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
03H05 Nonstandard models in mathematics
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[1] Weiss, G.M, Aspects and applications of random walk, (1994), North Holland Amsterdam
[2] Balescu, R, Statistical dynamics; matter out of equilibrium, (1997), Imperial College Press and World Scientific Singapore · Zbl 0997.82505
[3] Jumarie, G, A fokker – planck equation of fractional order with respect to time, J. math. phys., 33, 10, 3536-3542, (1992) · Zbl 0761.60071
[4] Ohdham, K.B; Spanier, J, The fractional calculus, (1974), Academic New York
[5] Jumarie, G, Maximum entropy, information without probability and complex fractals, (2000), Kluwer Academic Publishers Dordrecht
[6] Sainty, P, Construction of a complex-valued fractional Brownian motion of order, J. math. phys., 33, 9, 3128-3149, (1992) · Zbl 0762.60073
[7] Hochberg, G, Signed measure of probability, Ann. probab., 6, 3, 433-458, (1978) · Zbl 0378.60030
[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;(18):3563-7
[9] Barkai, E; Silbey, R.J, Fractional Kramers equation, J. phys. chem. B, 104, 3866-3874, (2000)
[10] Barkai E, Metzler R, Klafter J. From continuous time random walks to the fractional Fokker-Planck equation. Phys Rev E 61(1):132-8
[11] Hilfer, R; Anton, L, Phys. rev. E, 51, 848, (1994)
[12] El-Wakil, S.A; Zahran, M.A, Fractional integral representation of master equation, Chaos, solitons & fractals, 10, 1545, (1999) · Zbl 0988.82039
[13] Nottale, L, Fractal space-time and microphysics: towards a theory of scale relativity, (1993), World Scientific Singapore · Zbl 0789.58003
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