Showing papers by "Enzo Orsingher published in 2005"
TL;DR: In this article, the authors considered the planar random motion of a particle that moves with constant finite speed c and changes its direction 0 with uniform law in [0, 27r] and derived the explicit probability law f(x, y, t) of (X(t), Y(t)), t > 0.
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.
56 citations
TL;DR: In this paper, a logarithmic and exact small deviation asymptotics for the L 2 norm of certain Gaussian processes closely connected with a Wiener process are examined.
Abstract: We find some logarithmic and exact small deviation asymptotics for the L
2-norms of certain Gaussian processes closely connected with a Wiener process. In particular, processes obtained by centering and integrating Brownian motion and Brownian bridge are examined. Bibliography: 28 titles.
42 citations
TL;DR: In this paper, it was shown that the pseudoprocesses governed by heat-type equations of order n ⩾ 2 have a local time in zero whose distribution coincides with the folded fundamental solution of a fractional diffusion equation of order 2 (n - 1 ) / n, n ⊆ 2.
26 citations
TL;DR: In this paper, the explicit distribution of the position of randomly moving particles on the line and in the plane (with different velocities taken cyclically) by means of order statistics and by studying suitable problems of differential equations is derived.
Abstract: The aim of this paper is to derive the explicit distribution of the position of randomly moving particles on the line and in the plane (with different velocities taken cyclically) by means of order statistics and by studying suitable problems of differential equations. The two approaches are compared when both are applicable (case of the telegraph process). In some specific cases (alternating motions with skipping) it is possible to use the order statistics approach also to solve the equations governing the distribution. Finally, the approach based on order statistics is also applied in order to obtain the distribution of the position in the case of planar motion with three velocities conditioned on the number of changes of directions recorded.
19 citations
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01 Jan 2005
TL;DR: In this paper, the authors considered the planar random motion of a particle that moves with constant finite speed c and changes its direction 0 with uniform law in [0, 27r] and derived the explicit probability law f(x, y, t) of (X(t), Y(t)), t > 0.
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.
3 citations