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Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these equations provide probability density functions, evolving on time or variable in space, which are related to the class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions play a key role.
We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly,
Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations. As prototypes we study three variants of a generic time-fr
In this paper we review some general properties of probability distributions which exibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of such para
The article is devoted to the dynamics of systems with an anomalous scaling near a critical point. The fractional stochastic equation of a Lanvevin type with the $varphi^3$ nonlinearity is considered. By analogy with the model A the field theoretic m
We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski kinetic equations, which describe evolution of the systems influenced by stochastic forces distributed with stable probability laws. These equations generalize known kinetic equa