On a Telegraph-Type Equation with Non-Constant Coefficients Emerging in Randomly Accelerated Motions

In this paper 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 $X(t)=\int\limits_0^t V(s)ds)$ is analyzed). In particular, solutions of the form $(p,x,v)=ехр\{-2\lambda t\}q(x-vt,t^2/2)$ 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.