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 $\bold X(t), \; t>0,$ is represented by the stochastic motion of a particle moving with constant finite speed $c>0$ in the Euclidean plane $\Bbb R^2$ 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 $\bold X(t)$, 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 $\bold X(t)$ 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.