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 Non-Linear, Linear and Sublinear Death Processes

Abstract: This paper is devoted to the study of a fractional version of non-linear $\mathpzc{M}^ u(t)$, $t>0$, linear $M^ u (t)$, $t>0$ 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 $T_{2 u} (t)$, $t>0$. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation.

Angular processes related to Cauchy random walks

Abstract: We study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of non-linear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the $n$-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio.

Random motions in $\mathbb{R}^3$ with orthogonal directions

Abstract: This paper is devoted to the detailed analysis of three-dimensional motions in R 3 with orthogonal directions switching at Poisson times and moving with constant speed c > 0. The study of the random position at an arbitrary time t > 0 on the surface of the support, forming an octahedron S ct , is completely carried out on the edges E ct and faces F ct . In particular, the motion on the faces F ct is analysed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on F ct in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position p = p ( t, x, y, z ) inside S ct implied the derivation of a sixth-order partial differential equation governing p (expressed in terms of the products of three D’Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinates as the motion devolpes. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformely choosen at each Poisson event. If the switches are governed by homogeneus Poisson process many explicit results are obtained.

Random Motions at Finite Velocity on Non-Euclidean Spaces

Abstract: In this paper, random motions at finite velocity on the Poincaré half-plane and on the unit-radius sphere are studied. The moving particle at each Poisson event chooses a uniformly distributed direction independent of the previous evolution. This implies that the current distance d(P0,Pt) from the starting point P0 is obtained by applying the hyperbolic Carnot formula in the Poincaré half-plane and the spherical Carnot formula in the analysis of the motion on the sphere. We obtain explicit results of the conditional and unconditional mean distance in both cases. Some results for higher-order moments are also presented for a small number of changes of direction.

Fractional telegraph equation, Mittag-Leffler function, Hilfer derivative, Hadamard fractional derivative, Riesz-Feller space-fractional derivative

Abstract: In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.

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.