Stochastic dynamics of generalized planar random motions with orthogonal directions

Abstract

We study planar random motions with finite velocities, of norm $c>0$, along orthogonal directions and changing at the instants of occurrence of a non-homogeneous Poisson process with rate function $\lambda(t),\ t\ge0$. We focus on the distribution of the current position $\bigl(X(t), Y(t)\bigr),\ t\ge0$, in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases the process is located within the closed square $S_{ct}=\{(x,y)\in \mathbb{R}^2\,:\,|x|+|y|\le ct\}$ and we obtain the probability law inside $S_{ct}$, on the edge $\partial S_{ct}$ and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process $\bigl(X(t), Y(t)\bigr)$ is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to $\lambda(t)$ and velocity $c/2$. Finally, we extend the results to a wider class of orthogonal-type evolutions.