Random motions with finite velocity driven by a geometric counting process

Stochastic processes for the description of finite-velocity random motions have been widely studied during the last decades. Generally, they describe the motion of a particle moving with finite constant speed on the real line, or on more general domains with different directions switching at random times. The classical model of these motions is the one-dimensional (integrated) telegraph process in which the changes of directions of the two possible velocities are governed by the Poisson process [1, 2]. Later, the study of random motions in Rn has been performed by many authors over the years. For example, random motions with constant velocity, few directions in low dimension (2 or 3) and exponential switching times where the changes of speed follow specific rules (cyclic, orthogonal, etc) appeared in [3, 4, 5, 6]. Moreover, several generalization or modification of the telegraph process with underlying Poisson process have been proposed with the aim to construct solvable and tractable models of random motions. Motivated by all these studies and possible applications to real situations, such as in insurance, reliability, queues and biological environments, in this work we propose a new scheme for the underlying point process in suitable instances of the telegraph process [9]. Specifically, we refer to finite-velocity random motions in R^n, for n = 1,2, with 2 and 3 velocities alternating cyclically, respectively, where the number of motion switches along each possible direction follows a Geometric Counting Process (GCP) [7, 8]. In detail, we firstly analyze the process {(X(t), V (t))t ∈ R+}, with state space R × {v1 , v2}, which represents the motion of a particle on the real line, with alternating velocities v1 > v2. Such process has two components: a singular component, corresponding to the case in which there are no velocity switches, and an absolutely continuous component, related to the motion of the particle when the velocity changes at least once. Secondly, we study the process {(X(t), Y (t), V (t)), t ∈ R^+_0 }, with state-space R^2 × {v1 , v2,v3} , which describes a particle performing a planar motion with three specific directions. Once defined the region R(t) representing all the possible positions of the particle at a given time, the probability law shows that the distribution of this process is a mixture of two discrete components, describing the situations in which the particle is found on the boundary of the region, and an absolutely continuous part, related to the motion of the particle in the interior of R(t).This two-dimensional process well describes the motion of the particle in a turbulent medium, for example, in the presence of a vortex. The asymptotic probability law of the particle is also explored as a uniform distribution in case (i) and a three-peaked distribution in case (ii). Finally, we investigate a further exstension of the case (ii) when the particle moves cyclically in the three-dimensional space R3 with four possible directions. Also in this study we obtain the explicit probability distribution of the position of the moving particle. As example, we analyze a special case with three and four fixed cyclic directions. The results presented in this contribution are based on a joint work with Antonio Di Crescenzo and Verdiana Mustaro (cf. [9]) and on an ongoing joint research with Gabriella Verasani.