Marginals of the planar symmetric Markov random flight on long time intervals behave like the Goldstein-Kac telegraph process

Abstract

The planar symmetric Markov random flight $\mathbf{X}(t),t>0,$ is represented by the stochastic motion of a particle moving with constant finite speed $c>0$ in the Euclidean plane ${\mathbb{R}}^2$ and taking on its initial and each new directions at $\lambda$-Poisson ($\lambda >0$) distributed random time instants by choosing them at random according to the uniform distribution on the unit circumference. We consider the marginals of $\mathbf{X}(t)$, that is, the projection of this stochastic motion onto the axes. This projection onto the $x_1$-axis (respectively, onto the $x_2$-axis) represents a one-dimensional stochastic motion with random velocity $c\cos \alpha$ (respectively, with random velocity $c\sin \alpha$), where $\alpha$ is a random variable distributed uniformly on the interval $[0,2\pi )$. We prove that the density of the marginals of $\mathbf{X}(t)$ is asymptotically, as $t\to \infty$, equivalent to the density of the classical one-dimensional Goldstein-Kac telegraph process with parameters ($c,\lambda$). This unexpected and interesting result is confirmed by numerical calculations.

14 pages, 4 figures, 1 table