  We present a method that allows, under suitable equivariance and regularity
conditions, to determine the Poisson boundary of a diffusion starting from the
Poisson boundary of a sub-diffusion of the original one. We then give two
examples of application of this d\'evissage method. Namely, we first recover
the classical result that the Poisson boundary of Brownian motion on a
rotationally symmetric manifolds is generated by its escape angle, and we then
give an "elementary" probabilistic proof of the delicate result of [Bai08],
i.e. the determination of the Poisson boundary of the relativistic Brownian
motion in Minkowski space-time.
