Synthesis of scaling exponents: a fractional Levy model

Abstract

Mandelbrot introduced the concept of fractals to describe the non-Euclidean shape of many aspects of the natural world. In the time series context he proposed the use of fractional Brownian motion (fBm) to model non-negligible temporal persistence, the "Joseph Effect"; and Levy flights to quantify large discontinuities, the "Noah Effect". In space physics, both effects are manifested in the intermittency and long-range correlation which are by now well-established features of geomagnetic indices and their solar wind drivers. In order to capture and quantify the Noah and Joseph effects in one compact model we propose the application of the "bridging" fractional Levy motion (fLm) to space physics. We perform an initial evaluation of some previous scaling results in this paradigm, and show how fLm can model the previously observed exponents. We suggest some new directions for the future.

Please note that our identifications of FBM and "FLM" with fractional time derivatives, made just before equation 2), were in error. See http://www.bibsonomy.org/bibtex/29f4390d7d0965c4c7371b02ddd8f6c7b/statphys23