A cyclic random motion in $\mathbb{R}^3$ driven by geometric counting processes

TL;DR

This paper models a particle's cyclic random motion in three-dimensional space driven by geometric counting processes, deriving explicit probability laws, analyzing limiting behavior, and exploring first-passage-time problems.

Contribution

It provides explicit probability laws for the particle's position and investigates the process's limiting behavior and first-passage-time, introducing new analytical tools for cyclic random motions.

Findings

Explicit probability law of the particle's position is derived.

The process does not admit a stationary density.

Limiting behavior of the density is characterized as intensities tend to infinity.

Abstract

We consider the random motion of a particle that moves with constant velocity in $\mathbb{R}^3$. The particle can move along four directions with different speeds that are attained cyclically. It follows that the support of the stochastic process describing the particle's position at time $t$ is a tetrahedron. We assume that the sequence of sojourn times along each direction follows a geometric counting process (GCP). When the initial velocity is fixed, we obtain the explicit form of the probability law of the process $\boldsymbol{X}(t) = (X_1(t);X_2(t);X_3(t))$, $t > 0$, for the particle's position. We also investigate the limiting behavior of the related probability density when the intensities of the four GCPs tend to infinity. Furthermore, we show that the process does not admit a stationary density. Finally, we introduce the first-passage-time problem for the first component of $\boldsymbol{X}(t)$ through a constant positive boundary $\beta > 0$ providing the bases for future developments.