On a planar random motion with asymptotically correlated components

TL;DR

This paper analyzes a planar random motion with direction switches governed by a Poisson process, representing the components as telegraph processes, deriving exact distributions, and showing that the process converges to a correlated Brownian motion in the hydrodynamic limit.

Contribution

It provides a novel representation of the planar motion components as linear combinations of telegraph processes and characterizes their distributions and hydrodynamic limits.

Findings

Components are linear combinations of independent telegraph processes.

Exact distribution of the process is derived both inside and on the boundary of its support.

In the hydrodynamic limit, the process converges to a correlated planar Brownian motion.

Abstract

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.