arXiv subjects

Rudolf Gorenflo

Publications and source records attributed to Rudolf Gorenflo.

At least 19 recordsLinked to original sources

arXiv2018

The Mittag-Leffler function in the thinning theory for renewal processes

The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous random walk and consequently of fractional diffusion in a plenary lecture by the late Prof Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic theory of renewal processes including the standard and the fractional Poisson processes, here we have revised the original approach by Gnedenko and Kovalenko for convenience of the experts of stochastic processes who are not aware of the relevance of the Mittag-Leffler function

math.PR↗

arXiv2016

On the fractional Poisson process and the discretized stable subordinator

The fractional Poisson process and the Wright process (as discretization of the stable subordinator) along with their diffusion limits play eminent roles in theory and simulation of fractional diffusion processes. Here we have analyzed these two processes, concretely the corresponding counting number and Erlang processes, the latter being the processes inverse to the former. Furthermore we have obtained the diffusion limits of all these processes by well-scaled refinement of waiting times and jumps

math.PR↗

arXiv2014

Operator theoretic approach to the Caputo derivative and the fractional diffusion equations

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for continuously differentiable functions. Accordingly, in the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of the smooth functions that are too narrow. In this paper, we introduce a suitable definition of the Caputo derivative in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces.

math.AP↗

arXiv2014

Hilfer-Prabhakar Derivatives and Some Applications

We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

math.PR↗

arXiv2013

A Fractional Generalization of the Poisson Processes and Some of its Properties

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order $α~(0 < α\leq 1)$. A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors and sample path properties are studied. Finally we introduce the $q$-Mittag-Leffler process associated with the $q$-Mittag-Leffler distribution.

math.ST↗

arXiv2013

Laplace-Laplace analysis of the fractional Poisson process

We generate the fractional Poisson process by subordinating the standard Poisson process to the inverse stable subordinator. Our analysis is based on application of the Laplace transform with respect to both arguments of the evolving probability densities.

math.PR↗

arXiv2012

Parametric subordination in fractional diffusion processes

We consider simulation of spatially one-dimensional space-time fractional diffusion. Whereas in an earlier paper of ours we have developed the basic theory of what we call parametric subordination via three-fold splitting applied to continuous time random walk with subsequent passage to the diffusion limit, here we go the opposite way. Via Fourier-Laplace manipulations of the relevant fractional partial differential equation of evolution we obtain the subordination integral formula that teaches us how a particle path can be constructed by first generating the operational time from the physical time and then generating in operational time the spatial path. By inverting the generation of operational time from physical time we arrive at the method of parametric subordination. Due to the infinite divisibility of the stable subordinator, we can simulate particle paths by discretization where the generated points of a path are precise snapshots of a true path. By refining the discretization more and more fine details of a path become visible.

math.PR↗

arXiv2012

Random walk models approximating symmetric space-fractional diffusion processes

For the symmetric case of space-fractional diffusion processes (whose basic analytic theory has been developed in 1952 by Feller via inversion of Riesz potential operators) we present three random walk models discrete in space and time. We show that for properly scaled transition to vanishing space and time steps these models converge in distribution to the corresponding time-parameterized stable probability distribution. Finally, we analyze in detail a model, discrete in time but continuous in space, recently proposed by Chechkin and Gonchar. Concerning the inversion of the Riesz potential operator $I_0^α$ let us point out that its common hyper-singular integral representation fails for $α= 1. In our Section 2 we have shown that the corresponding hyper-singular representation for the inverse operator $D_0^α$ can be obtained also in the critical (often excluded) case $α= 1$, by analytic continuation

math.PR↗

arXiv2011

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; (rw_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.

math.PR↗

arXiv2010

Mittag-Leffler Waiting Time, Power Laws,Rarefaction, Continuous Time Random Walk, Diffusion Limit

We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic power law waiting time to the Mittag-Leffler waiting time distribution via rescaling and respeeding the clock of time. By a second respeeding (by rescaling the spatial variable) we obtain the diffusion limit of the continuous time random walk under power law regimes in time and in space. Finally, we exhibit the time-fractional drift process as a diffusion limit of the fractional Poisson process and as a subordinator for space-time fractional diffusion.

math.PR↗

arXiv2008

Fractional Calculus: Integral and Differential Equations of Fractional Order

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

math-ph↗

arXiv2008

Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects

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 CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.

cond-mat.stat-mech↗

arXiv2008

Some recent advances in theory and simulation of fractional diffusion processes

To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.

math.PR↗

arXiv2008

Time-fractional derivatives in relaxation processes: a tutorial survey

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity.

math-ph↗

arXiv2007

From Power Laws to Fractional Diffusion: the Direct Way

Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that, by a well-scaled passage to the diffusion limit, generalized diffusion processes, fractional in time as well as in space, are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure.

math.PR↗

arXiv2007

Some aspects of fractional diffusion equations of single and distributed order

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order $β\in (0,1)$. The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of sub-diffusion (the variance grows in time sub-linearly). A further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one. Then the fundamental solution is still a probability density of a non-Markovian process that, however, is no longer self-similar but exhibits a corresponding distribution of time-scales.

math-ph↗

arXiv2007

Time-fractional diffusion of distributed order

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 (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative except for the single order case where the two approaches turn to be equivalent. We consider with some detail two cases of order distribution: the double-order and the uniformly distributed order. For these cases we exhibit plots of the corresponding fundamental solutions and their variance, pointing out the remarkable difference between the two approaches for small and large times.

cond-mat.stat-mech↗

arXiv2007

The two forms of fractional relaxation of distributed order

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 (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost, in particular on the asymptotic behaviour of the fundamental solution at small and large times. We give an outline of the theory providing the general form of the solution in terms of an integral of Laplace type over a positive measure depending on the order-distribution. We consider with some detail two cases of fractional relaxation of distributed order: the double-order and the uniformly distributed order discussing the differences between the R-L and C approaches. For all the cases considered we exhibit plots of the solutions for moderate and large times.

cond-mat.stat-mech↗