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 nonhomogeneous Poisson process with rate function $$\lambda = \lambda (t),\ t\ge 0$$ . We focus on the distribution of the current position $$\bigl (X(t), Y(t)\bigr ),\ t\ge 0$$ , 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 (X, Y) is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to $$\lambda $$ and velocity c/2. Finally, we extend the results to a wider class of orthogonal-type evolutions.