Subordination pathways to fractional diffusion

The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and under power law regime is splitted into three distinct random walks: (rw ₁), a random walk along the line of natural time, happening in operational time; (w ₂), a random walk along the line of space, happening in operational time; (rw ₃), the inversion of (rw ₁), namely a random walk along the line of operational time, happening in natural time. Via the general integral equation of CTRW and appropriate rescaling, the transition to the diffusion limit is carried out for each of these three random walks. Combining the limits of (rw ₁) and (rw ₂) we get the method of parametric subordination for generating particle paths, whereas combination of (rw ₂) and (rw ₃) yields the subordination integral for the sojourn probability density in space - time fractional diffusion.     

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.     

Beyond the Poisson renewal process: A tutorial survey

After sketching the basic principles of renewal theory and recalling the classical Poisson process, we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of Mittag–Leffler and of Wright type. We compare these three processes with each other.     

Why the Mittag-Leffler Function Can Be Considered the Queen Function of the Fractional Calculus?

In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation We expect that in the next future this function will gain more credit in the science of complex systems. Finally, in an appendix we sketch some historical aspects related to the author’s acquaintance with this function.     

Energy propagation for dispersive waves in dissipative media

University of Bologna, Italy. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 7, pp. 650–664, July, 1993.     

Evolution equations for the probabilistic generalization of the Voigt profile function

The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random variables with Gaussian density (due to the Doppler effect) and Lorentzian density (due to the pressure effect). Since these densities belong to the class of symmetric Lévy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric Lévy densities. We study the case when the widths of the distributions considered depend on a scale factor τ that is representative of spatial inhomogeneity or temporal non-stationarity. The evolution equations for this probabilistic generalization of the Voigt function are here introduced and interpreted as generalized diffusion equations containing two Riesz space-fractional derivatives, thus classified as space-fractional diffusion equations of double order.     

The fundamental solutions for the fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0 < β ≤ 1/2 or 1/2 < β ≤ 1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and Signalling problems can be expressed in terms of an auxiliary function M(z;β), where z = |x|/t^β is the similarity variable. Such function is proved to be an entire function of Wright type.