5,440 research outputs found
Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion
In this paper we develop the spectral theory of the fractional Brownian
motion (fBm) using the ideas of Krein's work on continuous analogous of
orthogonal polynomials on the unit circle. We exhibit the functions which are
orthogonal with respect to the spectral measure of the fBm and obtain an
explicit reproducing kernel in the frequency domain. We use these results to
derive an extension of the classical Paley-Wiener expansion of the ordinary
Brownian motion to the fractional case.Comment: Published at http://dx.doi.org/10.1214/009117904000000955 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Geometrical enhancement of the electric field: Application of fractional calculus in nanoplasmonics
We developed an analytical approach, for a wave propagation in
metal-dielectric nanostructures in the quasi-static limit. This consideration
establishes a link between fractional geometry of the nanostructure and
fractional integro-differentiation. The method is based on fractional calculus
and permits to obtain analytical expressions for the electric field
enhancement.Comment: Published in EP
Continuous Time Random Walks in periodic systems: fluid limit and fractional differential equations on the circle
In this article, the continuous time random walk on the circle is studied. We
derive the corresponding generalized master equation and discuss the effects of
topology, especially important when Levy flights are allowed. Then, we work out
the fluid limit equation, formulated in terms of the periodic version of the
fractional Riemann-Liouville operators, for which we provide explicit
expressions. Finally, we compute the propagator in some simple cases. The
analysis presented herein should be relevant when investigating anomalous
transport phenomena in systems with periodic dimensions.Comment: 14 pages, 1 figure. References added. Published versio
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