Advances in Applied Probability

Probability distribution function for the Euclidean distance between two telegraph processes

Alexander D. Kolesnik

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Consider two independent Goldstein-Kac telegraph processes X1(t) and X2(t) on the real line R. The processes Xk(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.

Article information

Adv. in Appl. Probab., Volume 46, Number 4 (2014), 1172-1193.

First available in Project Euclid: 12 December 2014

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 82C70: Transport processes

Telegraph process telegraph equation persistent random walk probability distribution function Euclidean distance


Kolesnik, Alexander D. Probability distribution function for the Euclidean distance between two telegraph processes. Adv. in Appl. Probab. 46 (2014), no. 4, 1172--1193. doi:10.1239/aap/1418396248.

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