Fractal stochastic processes: Clusters and intermittancies

Abstract

Many scaling relations for complex systems in the physical sciences involve non-integer exponents. We list several examples all of which may not be well known. We interpret non-integer exponents as indicating singularities arising from a long tailed probability distribution governing the physical observables. If the first appropriate moment of the probability distribution diverges, then no scale exists in which to qauge measurements and phenomena occur on all scales. Self-similar fractals, non-differentiability, and also non-integer exponents will arise. Random walk examples are presented where the above characteristics appear simply and naturally. The analysis provides a generalization of Weierstrass' continuous, but nowhere differentiable function. Lastly, the Riemann Hypothesis is recast in a random walk framework.