Space inhomogeneous random motions of particles on the line and in the plane are considered in the paper.The changes of the movement direction are driven by a Poisson process.The particles are assumed to move according to a finite velocity field that depends on a spatial argument.The explicit distribution of particles is obtained in the paper for the case of dimension 1 in terms of characteristics of the governing equations.In the case of dimension 2, the distribution is obtained if a rectifying diffeomorphism exists.