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Kolesnik, Alexander D.

Probability distribution function for the Euclidean distance between two telegraph processes. (English) Zbl 1305.60103

Adv. Appl. Probab. 46, No. 4, 1172-1193 (2014).
Summary: Consider two independent Goldstein-Kac telegraph processes \(X_1(t)\) and \(X_2(t)\) on the real line \(\mathbb{R}\). 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 \(\mathbb{R}\) and are controlled by two independent homogeneous Poisson processes of rates \(\lambda_1>0\) and \(\lambda_2>0\), respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance \(\rho(t)= |X_1(t)-X_2(t)|\), \(t>0\), between these processes at an arbitrary time instant \(t>0\). Some numerical results are also presented.

Cited in 4 Documents

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60J60 Diffusion processes
60J65 Brownian motion
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics

Keywords:

telegraph process; telegraph equation; persistent random walk; probability distribution function; Euclidean distance
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Full Text: DOI arXiv Euclid

References:

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