2021/08/23 by Cinque, Fabrizio, Orsingher, Enzo
#60G50 #60K99 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2108.10027
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 λ(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 Sct=\(x,y)∈ ℝ2 : |x|+|y|≤ ct\ and we obtain the probability law inside Sct, on the edge ∂ Sct 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(t), Y(t)) is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to λ(t) and velocity c/2. Finally, we extend the results to a wider class of orthogonal-type evolutions.