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Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk
Rudolf Gorenflo, Francesco Mainardi
A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations…
Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects
Rudolf Gorenflo, Francesco Mainardi
We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional…
Probability distributions generated by fractional diffusion equations
Francesco Mainardi, Paolo Paradisi, Rudolf Gorenflo
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 deriv…
Discrete random walk models for space-time fractional diffusion
Rudolf Gorenflo, Francesco Mainardi, Daniele Moretti +2
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and relate…
Time-fractional diffusion of distributed order
Francesco Mainardi, Antonio Mura, Gianni Pagnini +1
The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (…
The two forms of fractional relaxation of distributed order
Francesco Mainardi, Antonio Mura, Rudolf Gorenflo +1
The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo…
Continuous time random walk and parametric subordination in fractional diffusion
Rudolf Gorenflo, Francesco Mainardi, Alessandro Vivoli
The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite s…
Anomalous waiting times in high-frequency financial data
Enrico Scalas, Rudolf Gorenflo, Francesco Mainardi +2
In high-frequency financial data not only returns, but also waiting times between consecutive trades are random variables. Therefore, it is possible to apply continuous-time random…
Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
A. V. Chechkin, R. Gorenflo, I. M. Sokolov
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,…
Non-Boltzmann Equilibrium Probability Densities for Non-Linear Lévy Oscillator
A. Chechkin, V. Gonchar, R. Gorenflo +2
We study, both analytically and by numerical modeling the equilibrium probability density function for an non-linear Lévy oscillator with the Lévy index α, 1 \leq α\leq 2, and the…
Fractional calculus and continuous-time finance II: the waiting-time distribution
Francesco Mainardi, Marco Raberto, Rudolf Gorenflo +1
We complement the theory of tick-by-tick dynamics of financial markets based on a Continuous-Time Random Walk (CTRW) model recently proposed by Scalas et al., and we point out its…
Discrete random walk models for symmetric Levy-Feller diffusion processes
Rudolf Gorenflo, Gianni De Fabritiis, Francesco Mainardi
We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index (), in the symmetric ca…