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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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Proceedings ArticleDOI
TL;DR: In this paper, the fractional Klein-Gordon equation involving fractional powers of the D'Alembert operator is reduced to a fractional hyper-Bessel-type equation.
Abstract: In this paper we discuss some exact results related to the fractional Klein--Gordon equation involving fractional powers of the D'Alembert operator. By means of a space-time transformation, we reduce the fractional Klein--Gordon equation to a fractional hyper-Bessel-type equation. We find an exact analytic solution by using the McBride theory of fractional powers of hyper-Bessel operators. A discussion of these results within the framework of linear dispersive wave equations is provided. We also present exact solutions of the fractional Klein-Gordon equation in the higher dimensional cases. Finally, we suggest a method of finding travelling wave solutions of the nonlinear fractional Klein-Gordon equation with power law nonlinearities.

1 citations

Journal ArticleDOI
TL;DR: In this paper, les distributions des supremums des champs aleatoires gaussiens Z(P)=∫G(P,P')dW(P'), ou dW =∫dW (P'), are compared.
Abstract: On compare les distributions des supremums des champs aleatoires gaussiens Z(P)=∫G(P,P')dW(P') et U(P)=∫dW(P'), ou dW est un champ de bruit blanc et G une fonction de reponse determinante particuliere, les integrales etant prises sur un cercle de rayon fixe. Les resultats permettent d'etablir des bornes superieures pour la loi du supremum de Z(P)

1 citations

Book ChapterDOI
29 May 2016
TL;DR: In this article, the authors considered random motions on the line and on the plane with space-varying velocities and analyzed the explicit distribution of the position of the moving particle.
Abstract: Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position \(\mathcal {T}(t)\), \(t>0\), of the moving particle. Also the case of a nonhomogeneous Poisson process (with rate \(\lambda = \lambda (t)\)) governing the changes of direction is analyzed in three specific cases. For the special case \(\lambda (t)= \alpha /t\), we obtain a random motion related to the Euler–Poisson–Darboux (EPD) equation which generalizes the well-known case treated, e.g., in (Foong, S.K., Van Kolck, U.: Poisson random walk for solving wave equations. Prog. Theor. Phys. 87(2), 285–292, 1992, [6], Garra, R., Orsingher, E.: Random flights related to the Euler-Poisson-Darboux equation. Markov Process. Relat. Fields 22, 87–110, 2016, [8], Rosencrans, S.I.: Diffusion transforms. J. Differ. Equ. 13, 457–467, 1973, [16]). A EPD-type fractional equation is also considered and a parabolic solution (which in dimension \(d=1\) has the structure of a probability density) is obtained. Planar random motions with space-varying velocities and infinite directions are finally analyzed in Sect. 5. We are able to present their explicit distributions, and for polynomial-type velocity structures we obtain the hyper- and hypoelliptic form of their support (of which we provide a picture).

1 citations

Posted Content
TL;DR: In this paper, the Fourier series of the signed density measures of pseudoprocesses on the line is derived for even-order and odd-order pseudo-processes.
Abstract: Pseudoprocesses, constructed by means of the solutions of higher-order heat-type equations have been developed by several authors and many related functionals have been analyzed by means of the Feynman-Kac functional or by means of the Spitzer identity. We here examine pseudoprocesses wrapped up on circles and derive their explicit signed density measures. We develop the Fourier series of their laws in the case of even-order and odd-order pseudoprocesses separately. We observe that circular even-order pseudoprocesses differ substantially from pseudoprocesses on the line because - for $t> \bar{t} > 0$, where $\bar{t}$ is a suitable $n$-dependent time value - they become real random processes. By composing the circular pseudoprocesses with positively-skewed stable processes we arrive at genuine circular processes whose distribution, in the form of Poisson kernels, is obtained. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal and therefore to the wrapped up law of Brownian motion. The last section of the paper is related to the circular Fresnel pseudoprocess and we examine its composition with stable subordinators. This leads to combinations of Poisson kernels.

1 citations

Posted Content
TL;DR: In this article, the authors presented the conditional distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals.
Abstract: In this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive initial velocity a reflection principle is proved to be valid while in the case of an initial leftward displacement the conditional distributions are perturbed by a positive probability of never visiting the half positive axis. Various relationships are established among the mentioned four classes of conditional distributions of the maximum. The unconditional distributions of the maximum of the telegraph process are obtained for positive and negative initial steps as well as their limiting behaviour. Furthermore the cumulative distributions and the general moments of the conditional maximum are presented.

1 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