Probability distribution function for the Euclidean distance between two telegraph processes

Abstract

Consider two independent Goldstein-Kac telegraph processes X₁(t) and X₂(t) on the real line R. The processes X_k(t), k = 1, 2, describe stochastic motions at finite constant velocities c₁ > 0 and c₂ > 0 that start at the initial time instant t = 0 from the origin of R and are controlled by two independent homogeneous Poisson processes of rates $λ₁ > 0 and $λ₂ > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X₁(t) - X₂(t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.