Probability Law For the Euclidean Distance Between Two Planar Random Flights

We consider two independent symmetric Markov random flights ${\mathbf{Z}}_1(t)$ and ${\mathbf{Z}}_2(t)$ performed by the particles that simultaneously start from the origin of the Euclidean plane ${\mathbb{R}}^2$ in random directions distributed uniformly on the unit circumference $S_1$ and move with constant finite velocities $c_1>0,c_2>0$, respectively. The new random directions are taking uniformly on $S_1$ at random time instants that form independent homogeneous Poisson flows of rates ${\lambda }_1>0,{\lambda }_2>0$. The probability distribution function of the Euclidean distance $$\rho (t)=\mathrm{\parallel }{\mathbf{Z}}_1(t)-{\mathbf{Z}}_2(t)\mathrm{\parallel },t>0,$$ between $\bold Z_1(t)$and$\bold Z_2(t)$atarbitrarytimeinstant$t>0$, is obtained.