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We consider a stochastic process $\{(\boldsymbol{X}(t),V(t)), t \geq 0\}$which describes a particle performing a minimal cyclic random motion withfinite velocities in $\mathbb{R}^3$. The particle can take four directionsmoving with different velocities $\vec{v}_j$, for $1 \leq j \leq4$, so that thediffusion region is a tetrahedron $\mathcal{T}(t)$. Moreover, we assume thatthe sequence of sojourn times along each velocity $\vec{v}_j$ follows ageometric counting process of intensity $\lambda_j$, $1 \leq j \leq4$. We firstdescribe the direction vectors $\vec{v}_j$ and the domain $\mathcal{T}(t)$;then, we obtain the closed-form expressions of the initial and absolutelycontinuous components of the probability law of the process when the startingvelocity is $\vec{v}_1$. We also investigate the limiting behavior of theprobability density of the process when the intensities $\lambda_j$ tend toinfinity. Finally, we introduce the first-passage time problem for the firstcomponent of $\boldsymbol{X}(t)$ through a constant positive boundary $\beta >0$ providing the bases for future developments.

A three-dimensional cyclic random motion with finite velocities driven by geometric counting processes