Cyclic planar random evolution with four directions

A four-direction cyclic random motion with constant finite speed $v$ in the plane $R^2$ driven by a homogeneous Poisson process of rate $\lambda>0$ is studied. A fourth-order hyperbolic equation with constant coefficients governing the transition law of the motion is obtained. A general solution of the Fourier transform of this equation is given. A special non-linear automodel substitution is found reducing the governing partial differential equation to the generalized fourth-order ordinary Bessel differential equation, and the fundamental system of its solutions is explicitly given.