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.

Damped wave equation

Abstract: We consider the planar random motion of a particle that moves with constant finite speed c and, at Poisson-distributed times, changes its direction 0 with uniform law in [0, 27r). This model represents the natural two-dimensional counterpart of the wellknown Goldstein-Kac telegraph process. For the particle's position (X (t), Y(t)), t > 0, we obtain the explicit conditional distribution when the number of changes of direction is fixed. From this, we derive the explicit probability law f(x, y, t) of (X(t), Y(t)) and show that the density p(x, y, t) of its absolutely continuous component is the fundamental solution to the planar wave equation with damping. We also show that, under the usual Kac condition on the velocity c and the intensity X of the Poisson process, the density p tends to the transition density of planar Brownian motion. Some discussions concerning the probabilistic structure of wave diffusion with damping are presented and some applications of the model are sketched.

The last zero-crossing of an iterated brownian motion with drift

Abstract: In this paper, we consider the iterated Brownian motion μ2μ1I(t)=B1μ1(|B2μ2(t)|) where Bjμj,j=1,2 are two independent Brownian motions with drift μj. Here, we study the last zero crossing before th...

Limiting distributions of randomly accelerated motions

Abstract: In this paper, the process {X(t); t>0}, representing the position of a uniformly accelerated particle (with Poisson-paced) changes of its acceleration, is studied. It is shown that the distribution ofX(t) (suitably normalized), conditionally on the numbern of changes of acceleration, tends in distribution to a normal variate asn goes to infinity. The asymptotic normality of the unconditional distribution ofX(t) for large values oft is also shown. The study of these limiting distributions is motivated by the difficulty of evaluating exactly the conditional and unconditional probability laws ofX(t). In fact, the results obtained in this paper permit us to give useful approximations of the probability distributions of the position of the particle.

Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres

Abstract: We consider diffusion processes \( {{\left( {{{{\underline{\mathrm{X}}}}_d}(t)} \right)}_{{t\geqslant 0}}} \) moving inside spheres \( S_R^d \) ⊂ ℝd and reflecting orthogonally on their surfaces. We present stochastic differential equations governing the reflecting diffusions and explicitly derive their kernels and distributions. Reflection is obtained by means of the inversion with respect to the sphere \( S_R^d \). The particular cases of Ornstein–Uhlenbeck process and Brownian motion are examined in detail.

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.