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 topic(s): Brownian motion & Fractional calculus. The author has an hindex of 30, co-authored 189 publication(s) receiving 3251 citation(s). Previous affiliations of Enzo Orsingher include University of Salerno.

##### Papers
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Journal ArticleDOI
TL;DR: In this paper, the fundamental solutions to time-fractional telegraph equations of order 2α were studied and the Fourier transform of the solutions for any α and the representation of their inverse, in terms of stable densities, was given.
Abstract: We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

245 citations

Journal ArticleDOI
Luisa Beghin
TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
Abstract: We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $u\in(0,1]$. For this process, denoted by $\mathcal{N}_ u(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_ u(t)= N(\mathcal{T}_{2 u}(t)),$ $t>0$. The time argument $\mathcal{T}_{2 u }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_ u.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $u\in(0,1]$ we show that the random position has a Brownian behavior (for $u =1/2$) or a cylindrical-wave structure (for $u =1$).

213 citations

Journal ArticleDOI
TL;DR: In this paper, the explicit form of the probability law and the associated flow function of a random motion governed by the telegraph equation are derived and connections of this law with the transition function of Brownian motion are explored.
Abstract: In this paper we derive the explicit form of the probability law and of the associated flow function of a random motion governed by the telegraph equation. Connections of this law with the transition function of Brownian motion are explored. Lower bounds for the distribution of its maximum are obtained and some particular distributions of its maximum, conditioned by the number of velocity reversals, are presented. Finally some versions of motion admitting annihilation are proven to be connected with Kirchoff's laws of electrical circuits.

124 citations

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed the solutions of fractional diffusion equations of order 0 < v ≤ 2 and interpreted them as densities of the composition of various types of stochastic processes.
Abstract: In this paper the solutions u ν = u ν (x, t) to fractional diffusion equations of order 0 < v ≤ 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order v = 1 2 n , n ≥ 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 v = 2 3 n , n ≥ 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 ν coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions u ν and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

115 citations

Journal ArticleDOI
TL;DR: In this paper, the authors introduce the space-fractional Poisson process whose state probabilities p, t, t > 0, � 2 (0,1), are governed by the equations (d/dt)pk(t) = � � (1 B)p � (t), where (B) is the fractional difference operator found in the study of time series analysis.
Abstract: In this paper we introduce the space-fractional Poisson process whose state probabilities p � (t), t > 0, � 2 (0,1], are governed by the equations (d/dt)pk(t) = � � (1 B)p � (t), where (1 B) � is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions p � (t), the probability generating functions G�(u,t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.

95 citations

##### Cited by
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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.

3,280 citations

Book ChapterDOI
01 Jan 2015

2,819 citations

Journal ArticleDOI

2,309 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.

1,794 citations