Fractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process. Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed. (c) 1997 American Institute of Physics.

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We propose a simple theory for the "universal" scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of inter-event times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforces the view that triggering of earthquakes by other earthquakes is a key physical mechanism to understand seismicity. PACS numbers: 91.30.Px ; 89.75.Da; Understanding the space-time-magnitude organization of earthquakes remains one of the major unsolved problem in the physics of the Earth. Earthquakes are characterized by a wealth of power laws, among them, (i) the Gutenberg-Richter distribution ∼ 1/E 1+β (with β ≈ 2/3) of earthquake energies E [1]; (ii) the Omori law ∼ 1/t p (with p ≈ 1 for large earthquakes) of the rate of aftershocks as a function of time t since a mainshock [2]; (iii) the productivity law ∼ E a (with a 2/3) giving the number of earthquakes triggered by an event of energy E [3]; (iv) the power law distribution ∼ 1/L 2 of fault lengths L [4]; (v) the fractal (and even probably multifractal [5]) structure of fault networks [6] and of the set of earthquake epicenters [7]. The quest to squeeze novel information from the observed properties of seismicity with ever new ways of looking at the data goes unabated in the hope of better understanding the physics of the complex solid Earth system. In this vein, from an analysis of the probability density functions (PDF) of waiting times between earthquakes in a hierarchy of spatial domain sizes and magnitudes in Southern California, Bak et al. discussed in 2002 a unified scaling law combining the Gutenberg-Richter law, the Omori law and the fractal distribution law in a single framework [8] (see also ref.[9] for a similar earlier study). This global approach was later refined and extended by the analysis of many different regions of the world by Corral, who proposed the existence of a universal scaling law for the PDF H(τ ) of recurrence times (or inter-event times) τ between earthquakes in a given region S [10, 11]:The remarkable finding is that the function f (x), which exhibit different power law regimes with cross-overs, is found almost the same for many different seismic regions, suggesting universality. The specificity of a given region seems to be completely captured solely by the average rate λ of observable events in that region, which fixes the only relevant characteristic time 1/λ.The common interpretation is that the scaling law (1) reveals a complex spatio-temporal organization of seismicity, which can be viewed as an intermittent flow of energy released within a self-organized (critical?) system [12], for which concepts and tools from the theory of critical phenomena...