  We recently introduced a new family of processes which describe particles
which only can move at the speed of light c in the ordinary 3D physical space.
The velocity, which randomly changes direction, can be represented as a point
on the surface of a sphere of radius c and its trajectories only may connect
the points of this variety. A process can be constructed both by considering
jumps from one point to another (velocity changes discontinuously) and by
continuous velocity trajectories on the surface. We followed this second new
strategy assuming that the velocity is described by a Wiener process (which is
isotropic only in the 'rest frame') on the surface of the sphere. Using both
Ito calculus and Lorentz boost rules, we succeed here in characterizing the
entire Lorentz-invariant family of processes. Moreover, we highlight and
describe the short-term ballistic behavior versus the long-term diffusive
behavior of the particles in the 3D physical space.
