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Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Part of: Special processes Markov processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  22 February 2016

Alexander D. Kolesnik
Alexander D. Kolesnik*
Affiliation:
Academy of Sciences of Moldova
*
∗ Postal address: Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Academy Street 5, Kishinev 2028, Moldova. Email address: kolesnik@math.md
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Abstract

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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 λ1 > 0 and λ2 > 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.

Keywords

Telegraph processtelegraph equationpersistent random walkprobability distribution functionEuclidean distance

MSC classification

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

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1172 - 1193
Copyright
© Applied Probability Trust 

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