Enzo Orsingher

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.

Fractional Klein–Gordon Equations and Related Stochastic Processes

Abstract: This paper presents finite-velocity random motions driven by fractional Klein–Gordon equations of order \(\alpha \in (0,1]\). A key tool in the analysis is played by the McBride’s theory which converts fractional hyper-Bessel operators into Erdelyi–Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein–Gordon equation. The distribution of the fractional telegraph process for \(\alpha = 1\) coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein–Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations \(B(n,\alpha )\) (with uniformly-distributed orientation) are considered. The parameter \(n\) of \(B(n,\alpha )\) is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of direction. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to \(N\)-dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler–Poisson–Darboux equation to which our theory applies.

State-dependent fractional point processes

Abstract: In this paper we analyse the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < ν k ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of p k ν k (t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on ν k differs from that constructed from the fractional state equations (in the case of ν k = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

On a fractional linear birth--death process

Abstract: In this paper, we introduce and examine a fractional linear birth--death process $N_{ u}(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^{ u}(t)$, $t>0$, $k\geq0$. We present a subordination relationship connecting $N_{ u}(t)$, $t>0$, with the classical birth--death process $N(t)$, $t>0$, by means of the time process $T_{2 u}(t)$, $t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^{ u}(t)$ and the state probabilities $p_k^{ u}(t)$, $t>0$, $k\geq1$, in the three relevant cases $\lambda>\mu$, $\lambda<\mu$, $\lambda=\mu$ (where $\lambda$ and $\mu$ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth--death process with the fractional pure birth process. Finally, the mean values $\mathbb{E}N_{ u}(t)$ and $\operatorname {\mathbb{V}ar}N_{ u}(t)$ are derived and analyzed.

Fractional Non-Linear, Linear and Sublinear Death Processes

Abstract: This paper is devoted to the study of a fractional version of non-linear , t>0, linear M ν(t), t>0 and sublinear $\mathfrak{M}^{ u}(t)$ , t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T 2ν(t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Table Of Integrals Series And Products

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Lévy processes and infinitely divisible distributions

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.