Stochastic Dynamics of Generalized Planar Random Motions with Orthogonal Directions

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 >= 0. We focus on the distribution of the current position (X(t), Y(t)), t >= 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) is an element of R-2 : |x| + |y| <= ct} and we obtain the probability law inside S-ct, on the edge & part;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.