On a Planar Random Motion with Asymptotically Correlated Components

We study a planar random motion \(\big (X(t),\,Y(t)\big )\) with orthogonal directions, where the direction switches are governed by a homogeneous Poisson process. At each Poisson event, the moving particle turns clockwise or counterclockwise according to a rule which depends on the current direction. We prove that the components of the vector \(\big (X(t),\,Y(t)\big )\) can be represented as linear combinations of two independent telegraph processes with different intensities. The exact distribution of \(\big (X(t),\,Y(t)\big )\) is then obtained both in the interior of the support and on its boundary, where a singular component is present. We show that, in the hydrodynamic limit, the process behaves as a planar Brownian motion with correlated components. The distribution of the time spent by the process moving vertically is then studied. We obtain its exact distribution and discuss its hydrodynamic limit. In particular, in the limiting case, the process \(\big (X(t),\,Y(t)\big )\) spends half of the time moving vertically.