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.

Stochastic motions driven by wave equations

Abstract: In this paper we present some random motions whose probability law is a solution of the telegraph equation. We give the explicit form of the flow function and of the probability law whose convergence to the Gaussian distribution is also discussed. A planar version of motion governed by the equation of damped vibrations of membranes is also presented. Finally the connection of this motions with Maxwell equations of electrodynamics is analysed.

Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

Abstract: Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The analysis of the fractional version (of order $ u$) of the Fresnel equation is also performed and, in detail, some specific cases, like $ u=1/2$, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process $F(t)$, $t>0$ with real sign-varying density is constructed and some of its properties examined. The equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. The composition of F with reflecting Brownian motion $B$ yields the law of biquadratic heat equation while the composition of $F$ with the first passage time $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.

Some results on random flights

Abstract: Abstract. We consider in this paper random flights in d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on spheres S 1 . For the position (X1(t), ..., Xd(t)) we obtain the conditional characteristic function E e d k=1 αkXk(t) | N(t) = n and related density pn(x1, ..., xd; t) in terms of (n + 1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d = 2 and d = 4. In these two cases also the unconditional distribution is determined in explicit form. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d = 2. The related motion with random velocity in 3 is analyzed and its distribution derived.

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.