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From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion
1.A. Einstein, Investigations on the Theory of Brownian Movement (Dover, New York, 1956).
6.A. Blumen, J. Klafter, and G. Zumofen in Optical Spectroscopy of Glasses, edited by I. Zschokke (Reidel, Dordrecht, 1986) pp. 199–265.
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9.G. G. Infante López, H. Hermanns and J.-P. Katoen, Lect. Notes Comput. Sci. 2165, 57 (2001);
9.see also N. Limnios, Commun. Stat: Theory Meth. 33, 3, xiii (2004).
10.A. I. Saichev, and S. G. Utkin, Modern Problems of Statistical Physics 1, 5 (2002).
11.Here and further on, the function is understood in the sense it usually has in the theory of Laplace transform, it is, namely, a distribution for which ; more carefully one should write , with .
16.The function is a Laplace transform of a pdf and therefore is completely monotonous. This means that it is positive and monotonously nongrowing. Moreover, so that is monotonously nondecaying and . For all distributions except for it is monotonously growing and thus possesses no zeros except for one for (which corresponds to ). Therefore all divisions or multiplications discussed in the text are essentially harmless operations.
17.I. M. Sokolov, A. V. Chechkin, and J. Klafter, Acta Phys. Pol. B, 35, 1323 (2004).
18.M. Caputo, Elasticità e Dissipazione (Zanichelli Printer, Bologna, 1969).
24.A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Yu. Gonchar, Fractional Calculus and Applied Analysis 6, 259 (2003).
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