Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Abstract

Consider two independent Goldstein-Kac telegraph processes X 1 (t) and X 2 (t) on the real line R. The processes X k (t), k = 1, 2, describe stochastic motions at finite constant velocities c 1 > 0 and c 2 > 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 λ 1 > 0 and λ 2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X 1 (t) − X 2 (t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.

Probability Law for the Euclidean Distance Between Two Planar Random Flights

“…It is clear that 0 < ρ(t) < (c 1 + c 2 )t with probability 1 for any t > 0, that is, the open interval (0, (c 1 + c 2 )t) is the support of the distribution of process ρ(t). Note that, in contrast to the one-dimensional case (see [2]), the distribution of the Euclidean distance (4.1) is absolutely continuous in the interval (0, (c 1 + c 2 )t) and does not contain any singular component. This means that probability distribution function (4.2) is continuous for r ∈ R and does not have any jumps.…”

“…The method of obtaining a formula for Φ(r, t) is different from that used in the onedimensional case (see [2]). While in [2] the method is based on evaluating the probability the particle to be located in a r-neighbourhood of the other one, in the multidimensional case such approach is impracticable.…”

“…However, despite the importance of this problem, it has not almost been studied in the literature. The only recent result [2] yields a closed-form formula for the probability distribution function of the Euclidean distance between two independent Goldstein-Kac telegraph processes. To the best of our knowledge, the multidimensional counterparts of this problem were not examined yet.…”

Linear combinations of the telegraph random processes driven by partial differential equations

“…It was also shown that the shifted time derivative of the transition density satisfies the telegraph equation with doubled parameters 2c and 2λ. A functional relation connecting the distributions of the difference of two independent telegraph processes with arbitrary parameters and of the Euclidean distance between them, was given in [18,Remark 4.4].…”

“…However, to the best of the author's knowledge, there exist only a few works where a system of several telegraph processes is considered. A closed-form expression for the probability distribution function of the Euclidean distance between two independent telegraph processes with arbitrary parameters was derived in [18]. The explicit probability distribution for the sum of two independent telegraph processes with the same parameters, both starting from the origin of R, was obtained in [17].…”