caa a22 4500 467891974 CHVBK 20180406152751.0 cr unu---uuuuu 170328e20060501xx s 000 0 eng 10.1007/s10955-006-9074-2 doi (NATIONALLICENCE)springer-10.1007/s10955-006-9074-2 Anomalous Diffusion Index for Lévy Motions [Elektronische Daten] [Chang Dorea, Ary Medino] In modelling complex systems as real diffusion processes it is common to analyse its diffusive regime through the study of approximating sequences of random walks. For the partial sums $$S_n=\xi_1 + \xi_2 + \ldots + \xi_n$$ one considers the approximating sequence of processes $$X^{(n)}(t)= a_n (S_{[k_nt]}-b_n)$$ . Then, under sufficient smoothness requirements we have the convergence to the desired diffusion, $$X^{(n)}(t) \to X(t)$$ . A key assumption usually presumed is the finiteness of the second moment, and, hence the validity of the Central Limit Theorem. Under anomalous diffusive regime the asymptotic behavior of S n may well be non-Gaussian and $$n^{-1}E(S^2_n) \to\infty$$ . Such random walks have been referred by physicists as Lévy motions or Lévy flights. In this work, we introduce an alternative notion to classify these regimes, the diffusion index $$\gamma_X$$ . For some $$\gamma^0_X$$ properly chosen let $$\gamma_X=\inf \{ \gamma:0 < \gamma\leq\gamma^0_X,\limsup_{t\to \infty}t^{-1}E|X(t)|^{1/\gamma} < \infty \}$$ . Relationship between $$\gamma_X$$ , the infinitesimal diffusion coefficients and the diffusion constant will be explored. Illustrative examples as well as estimates, based on extreme order statistics, for $$\gamma_X$$ will also be presented. Springer Science + Business Media, Inc., 2006 Lévy motions nationallicence anomalous diffusion nationallicence Dorea Chang Departamento de Matemática, Universidade de Brasilia, 70910-900, Brasilia-DF, Brazil aut Medino Ary Departamento de Matemática, Universidade de Brasilia, 70910-900, Brasilia-DF, Brazil aut Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 123/3(2006-05-01), 685-698 0022-4715 123:3<685 2006 123 10955 https://doi.org/10.1007/s10955-006-9074-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-006-9074-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Dorea Chang Departamento de Matemática, Universidade de Brasilia, 70910-900, Brasilia-DF, Brazil aut NATIONALLICENCE 700 1- Medino Ary Departamento de Matemática, Universidade de Brasilia, 70910-900, Brasilia-DF, Brazil aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 123/3(2006-05-01), 685-698 0022-4715 123:3<685 2006 123 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer