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On a telegraph-type equation with non-constant coefficients emerging in randomly accelerated motions. (English. Russian original) Zbl 0917.35068
Probl. Inf. Transm. 30, No. 2, 177-182 (1994); translation from Probl. Peredachi Inf. 30, No. 99-103 (1994).
Summary: We analyze the random motion of a particle whose acceleration is the two-valued telegraph process \(\{A(t), t\geq 0\}\). We derive the third-order, hyperbolic partial differential equation governing the probability law \(p= p(x,v,t)\) of the Markov vector-valued process \(\{V(t), X(t), t\geq 0\}\) (\(V\) is obtained by integrating the two-valued telegraph process and \(x(t)= \int^t_0 V(s)ds\) is analyzed). In particular, solutions of the form \(p(x,v,t)= e^{-2\lambda t}q(x- vt, t^2/2)\) are taken into account. The general solution (in terms of the double-Fourier transform) of the equation governing \(q\) is presented, and some of its properties investigated.

35L25 Higher-order hyperbolic equations