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Author

Enzo Orsingher

Other affiliations: University of Salerno
Bio: Enzo Orsingher is an academic researcher from Sapienza University of Rome. The author has contributed to research in topics: Brownian motion & Fractional calculus. The author has an hindex of 30, co-authored 189 publications receiving 3251 citations. Previous affiliations of Enzo Orsingher include University of Salerno.


Papers
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Journal ArticleDOI
TL;DR: In this article, the authors considered some fractional extensions of the recursive differential equation governing the Poisson process, i.e. the generalized Mittag-Leffler functions.
Abstract: We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. $\partial_tp_k(t)=-\lambda(p_k(t)-p_{k-1}(t))$, $k\geq0$, $t>0$ by introducing fractional time-derivatives of order $ u,2 u,\ldots,n u$. We show that the so-called "Generalized Mittag-Leffler functions" $E_{\alpha,\beta^k}(x)$, $x\in\mathbb{R}$ (introduced by Prabhakar [24] )arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for $t\to\infty$. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter $ u$ varying in $(0,1]$. For integer values of $ u$, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.

92 citations

Journal ArticleDOI
TL;DR: In this article, the authors analyzed and interpreted the densities of the composition of various types of stochastic processes and showed that the solutions of fractional diffusion equations correspond to the distribution of the $n$-times iterated Brownian motion.
Abstract: In this paper the solutions $u_{ u}=u_{ u}(x,t)$ to fractional diffusion equations of order $0< u \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $ u =\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{{1/2^n}}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $ u =\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{ u}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{ u}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

92 citations

Journal ArticleDOI
Abstract: The space-fractional telegraph equation is analyzed and the Fourier transform of its fundamental solution is obtained and discussed. A symmetric process with discontinuous trajectories, whose transition function satisfies the space-fractional telegraph equation, is presented. Its limiting behaviour and the connection with symmetric stable processes is also examined.

83 citations

Journal ArticleDOI
TL;DR: In this article, the conditional characteristic function of the position of a particle after n changes of direction was obtained from this characteristic function and the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions.
Abstract: We consider in this paper random flights in ℝd performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S1d. We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ3 are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2.

82 citations

Journal ArticleDOI
TL;DR: This paper finds conditions of existence of the means and bounds for their values, involving also the truncated BDP XN and presents some examples where these bounds are used in order to approximate the double mean.
Abstract: In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t? 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP XN. Finally we present some examples where these bounds are used in order to approximate the double mean.

73 citations


Cited by
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Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

01 Jan 2016
TL;DR: The table of integrals series and products is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can get it instantly.
Abstract: Thank you very much for downloading table of integrals series and products. Maybe you have knowledge that, people have look hundreds times for their chosen books like this table of integrals series and products, but end up in harmful downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they cope with some harmful virus inside their laptop. table of integrals series and products is available in our book collection an online access to it is set as public so you can get it instantly. Our book servers saves in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the table of integrals series and products is universally compatible with any devices to read.

4,085 citations

Book ChapterDOI
01 Jan 2015

3,828 citations

Book
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

1,957 citations