On multifractality and fractional derivatives

U. Frisch and T. Matsumoto

Papers from arXiv.org

Abstract: It is shown phenomenologically that the fractional derivative $\xi=D^\alpha u$ of order $\alpha$ of a multifractal function has a power-law tail $\propto |\xi| ^{-p_\star}$ in its cumulative probability, for a suitable range of $\alpha$'s. The exponent is determined by the condition $\zeta_{p_\star} = \alpha p_\star$, where $\zeta_p$ is the exponent of the structure function of order $p$. A detailed study is made for the case of random multiplicative processes (Benzi {\it et al.} 1993 Physica D {\bf 65}: 352) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, which involves the departure from straightness of the $\zeta_p$ graph, for the presence of power-law tails when there is only a limited range over which the data possess scaling properties (e.g. because of the presence of a viscous cutoff). The method is also applied to wind tunnel data and financial data.

Date: 2001-07, Revised 2002-06
References: Add references at CitEc
Citations:

Published in J. Stat. Phys. 108, 1181-1202 (2002)

Downloads: (external link)
http://arxiv.org/pdf/nlin/0107057 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:nlin/0107057

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().