Planar random motions with drift (Q1872621)

The paper investigates the two-dimensional distribution of a particle moving randomly in the plane with finite velocity. The particle can move in four different directions with different velocities for each direction. Switching occurs accordingly to a homogeneous Poisson process, and each of the two neighboring directions is chosen with probability 1/2. Reflection is thus not possible. Firstly, the governing equations are derived for the case with four orthogonal directions. Secondly, an explicit solution in terms of Bessel functions is given for the case where the planar motion can be represented as a linear combination of two one-dimensional telegraph processes. Thirdly, for the case with three directions collinear with the coordinate axis and the fourth direction and velocity given as a function of the three other velocities, the governing equation is solved analytically without using the representation via the telegraph processes.