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.

On the solutions of linear odd-order heat-type equations with random initial conditions

Abstract: In this paper odd-order heat-type equations with different random initial conditions are examined. In particular, we give rigorous conditions for the existence of the solutions in the case where the initial condition is represented by a strictly ϕ –subGaussian harmonizable process η = η (x). Also the case where η is represented by a stochastic integral with respect to a process with independent increment is studied.

Some probabilistic properties of fractional point processes

Abstract: In this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernstein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form where are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process.

Gaussian limiting behavior of the rescaled solution to the linear Korteweg-de vries equation with random initial conditions

Abstract: We analyze the asymptotic behavior of the rescaled solution to the linear Korteweg–de Vries equation when the initial conditions are supposed to be random and weakly dependent. By means of the method of moments we prove the Gaussianity of the limiting process and we present its correlation function. The same technique is applied to the analysis of another third-order heat-type equation.

Composition of Processes and Related Partial Differential Equations

Abstract: In this paper various types of compositions involving independent fractional Brownian motions \(B^{j}_{H_{j}}(t)\), t>0, j=1,2, are examined.

Bose-Einstein-type statistics, order statistics and planar random motions with three directions

Abstract: In this paper we study different types of planar random motions (performed with constant velocity) with three directions, defined by the vectors dj = (cos (2πj/3), sin (2πj/3)) for j = 0, 1. 2, changing at Poisson-paced times. We examine the cyclic motion (where the change of direction is deterministic), the completely uniform motion (where at each Poisson event each direction can be taken with probability 1 3) and the symmetrically deviating case (where the particle can choose all directions except that taken before the Poisson event). For each of the above random motions we derive the explicit distribution of the position of the particle, by using an approach based on order statistics. We prove that the densities obtained are solutions of the partial differential equations governing the processes. We are also able to give the explicit distributions on the boundary and, for the case of the symmetrically deviating motion, we can write it as the distribution of a telegraph process. For the symmetrically deviating motion we use a generalization of the Bose-Einstein statistics in order to determine the distribution of the triple (No, N 1 , N 2 ) (conditional on N(t) = k, with N 0 + N 1 + N 2 = N(t) + 1, where N(t) is the number of Poisson events in [0, t]), where Nj denotes the number of times the direction dj (j = 0, 1, 2) is taken. Possible extensions to four directions or more are briefly considered.

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.