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

Abstract

We consider a stochastic process $\{(\boldsymbol{X}(t),V(t)), t \geq 0\}$ which describes a particle performing a minimal cyclic random motion with finite velocities in $\mathbb{R}^3$. The particle can take four directions moving with different velocities $\vec{v}_j$, for $1 \leq j \leq4$, so that the diffusion region is a tetrahedron $\mathcal{T}(t)$. Moreover, we assume that the sequence of sojourn times along each velocity $\vec{v}_j$ follows a geometric counting process of intensity $\lambda_j$, $1 \leq j \leq4$. We first describe the direction vectors $\vec{v}_j$ and the domain $\mathcal{T}(t)$; then, we obtain the closed-form expressions of the initial and absolutely continuous components of the probability law of the process when the starting velocity is $\vec{v}_1$. We also investigate the limiting behavior of the probability density of the process when the intensities $\lambda_j$ tend to infinity. 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