Showing papers by "Enzo Orsingher published in 2004"
TL;DR: In this paper, the fundamental solutions to time-fractional telegraph equations of order 2α were studied and the Fourier transform of the solutions for any α and the representation of their inverse, in terms of stable densities, was given.
Abstract: We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α=1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.
264 citations
TL;DR: In this article, the authors studied different types of planar random motions with three directions and derived the explicit distribution of the position of the particle, by using an approach based on order statistics, and proved that the densities obtained are solutions of the partial differential equations governing the processes.
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.
15 citations