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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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20 Apr 2022
TL;DR: In this article , the authors study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators and obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent ν > 1 and skewness parameter β with 0 < | β | < 1 .
Abstract: In this paper we study pseudo-processes related to odd-order heat-type equations composed with Lévy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be represented as an expectation of damped oscillations with generalized gamma distributed parameters. This stochastic representation also arises as the solution to a fractional diffusion equation, involving a higher-order Riesz-Feller operator, which generalizes the odd-order heat-type equation. We then prove that, if the stable subordinator has a suitable exponent, the time-changed pseudo-process becomes a genuine Lévy stable process. This result permits us to obtain a power series representation for the probability density function of an arbitrary asymmetric stable process of exponent ν > 1 and skewness parameter β , with 0 < | β | < 1 . The methods we use in order to carry out our analysis are based on the study of a fractional Airy function which emerges in the investigation of the higher-order Riesz-Feller operator.

2 citations

Posted Content
TL;DR: In this paper, the first hitting times of generalized Poisson processes were studied and the hitting probabilities of these processes were explicitly obtained and analyzed for the space-fractional Poisson process.
Abstract: This paper studies the first hitting times of generalized Poisson processes $N^f(t)$, related to Bernstein functions $f$. For the space-fractional Poisson processes, $N^\alpha(t)$, $t>0$ (corresponding to $f= x^\alpha$), the hitting probabilities $P\{T_k^\alpha<\infty\}$ are explicitly obtained and analyzed. The processes $N^f(t)$ are time-changed Poisson processes $N(H^f(t))$ with subordinators $H^f(t)$ and here we study $N\left(\sum_{j=1}^n H^{f_j}(t)\right)$ and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form $N(|\mathcal{G}_{H, u}(t)|)$ where $\mathcal{G}_{H, u}(t)$ are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space-time Poisson process is no longer a renewal process.

2 citations

01 Jan 1986
TL;DR: In this article, a planar random motion governed by the two-dimensional telegraph equation is presented and it is proved that the particle performing motion is at any time t within a circle centred at the starting point and with radius ct/V2.
Abstract: In this paper a planar random motion governed by the two-dimensional telegraph equation is presented. It is proved that the particle performing motion is at any time t within a circle centred at the starting point and with radius ct//V2. The explicit density of the particle position is obtained. Results concerning the trend of motion are also given.

2 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the case where a particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless, and they show that their process converges to a Markovian process, namely a random flight on the hyperbolic manifold.
Abstract: We consider the motion of a particle along the geodesic lines of the Poincare half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.

2 citations

Posted Content
TL;DR: In this paper, the authors studied random flights in R^d with displacements possessing Dirichlet distributions of two different types and uniformly oriented, and proved that the distributions of the point X(t) and Y(t), t \geq 0, performing the random flights are related to Klein-Gordon type partial differential equations.
Abstract: In this paper we study random flights in R^d with displacements possessing Dirichlet distributions of two different types and uniformly oriented. The randomization of the number of displacements has the form of a generalized Poisson process whose parameters depend on the dimension d. We prove that the distributions of the point X(t) and Y(t), t \geq 0, performing the random flights (with the first and second form of Dirichlet intertimes) are related to Klein-Gordon-type partial differential equations. Our analysis is based on McBride theory of integer powers of hyper-Bessel operators. A special attention is devoted to the three-dimensional case where we are able to obtain the explicit form of the equations governing the law of X(t) and Y(t). In particular we show that that the distribution of Y(t) satisfies a telegraph-type equation with time-varying coefficients.

2 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