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.

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

Abstract: We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ℝ d , like, for example, (B 1(|C(t)|),…, B d (|C(t)|)) and (C 1(|C(t)|),…, C d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B 1(|B 2(…|B n+1(t)|…)|)|) are governed by fractional diffusion equations.

Analysis of a Finite-Velocity Planar Random Motion with Reflection

Abstract: A four-direction planar random motion with finite velocity and with possible reflections at Poisson-paced events is examined. We obtain the equations governing the distributions within the diffusion set $Q_t$ and the equations directing the singular components of the distributions. The distributions on the edge of $Q_t$ and its diagonals are explicitly obtained.

Pseudoprocesses related to space-fractional higher-order heat-type equations

Abstract: In this article, we construct pseudo random walks (symmetric and asymmetric) that converge in law to compositions of pseudoprocesses stopped at stable subordinators. We find the higher-order space-fractional heat-type equations whose fundamental solutions coincide with the law of the limiting pseudoprocesses. The fractional equations involve either Riesz operators or their Feller asymmetric counterparts. The main result of this article is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order γ > 2. The classical pseudoprocesses are very special cases of those investigated here.

On the distribution of the maximum of the telegraph process

Abstract: In this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive initial velocity a reflection principle is proved to be valid while in the case of an initial leftward displacement the conditional distributions are perturbed by a positive probability of never visiting the half positive axis. Various relationships are established among the mentioned four classes of conditional distributions of the maximum. The unconditional distributions of the maximum of the telegraph process are obtained for positive and negative initial steps as well as their limiting behaviour. Furthermore the cumulative distributions and the general moments of the conditional maximum are presented.

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.