A planar random motion with an infinite number of directions controlled by the 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 θ with uniform law in [0, 2π). This model represents the natural two-dimensional counterpart of the well-known 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 λ 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.

Stochastic velocity motions and processes with random time

“…Theorem 2.4 permits us to point out the connection between X 0 (t), X 1 (t), and the random flights studied in Stadje (1987), Kolesnik and Orsingher (2005), and De Gregorio and Orsingher (2007). Indeed, by setting ν = 0 in f ν (θ), we again obtain the uniform distribution on a unit semicircle, and (as expected) X 0 (t) represents the projection onto the x-axis of a planar random flight.…”

“…These results permit us to point out the relationship between X 0 (t), t > 0, and X 1 (t), t > 0, and the random flights studied by several authors, such as Stadje (1987Stadje ( ), (1989, Kolesnik and Orsingher (2005), De Gregorio and Orsingher (2006), Kolesnik (2006), and Orsingher and De Gregorio (2007). A random flight is a continuoustime random walk defined similarly to X ν (t), but with its direction chosen uniformly on an hypersphere.…”

“…In other words, X ν (t), t > 0, defines a whole class of random motions indexed by the parameter ν, namely the level of friction. For ν = 0, we obtain again the uniform distribution on the semicircle with radius 1 and X 0 (t) is exactly the x-component of a planar random flight studied in, for example, Stadje (1987) and Kolesnik and Orsingher (2005). Our first result concerns the conditional characteristic function of X ν (t), t > 0, for a fixed number of Poisson events during the time interval [0, t].…”

“…and, by means of the approach developed in Kolesnik and Orsingher (2005) (see the proof of Theorem 1), we observe that the n-fold integral…”

Random Motion with Uniformly Distributed Directions and Random Velocity

“…However, it has been observed in two-dimensional cases [14] and in three-dimensional cases [2] for constant velocity for the absolute continuous part of the distribution of the particle position. In this paper we also include a four-dimensional case where this phenomenon is present as well.…”