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Zbl 1195.37054
Jumarie, Guy
An approach via fractional analysis to non-linearity induced by coarse-graining in space.
(English)
[J] Nonlinear Anal., Real World Appl. 11, No. 1, 535-546 (2010). ISSN 1468-1218

Loosely speaking, a coarse-grained space is a space in which the generic point is not infinitely thin, but rather has a thickness; and here this feature is modeled as a space in which the generic increment is not $dx$, but rather $(dx)^\alpha, 0<\alpha<1$. The purpose of the article is to analyse the non-linearity induced by this coarse-graining effect. This approach via $(dx)^\alpha$ leads us to the use of fractional analysis which thus provides models in the form of nonlinear differential equations of fractional order. Two illustrative examples are considered. In the first one, one shows that a particle which has a Gaussian white noise in a coarse-grained space exhibits a trajectory which looks like a generalized fractional Brownian motion. The second example shows how a simple one-dimensional linear dynamics is converted into a non-linear system of fractional order.
[Fuhua Ling (Milpitas)]
MSC 2000:
*37L99 Infinite-dimensional dissipative dynamical systems
60J65 Brownian motion
60H40 White noise theory

Keywords: coarse-grained space; fractional calculus; fractional Taylor's series; modified Riemann-Liouville derivative; fractional Brownian motion

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