  In the setting of finite reflection groups, we prove that the projection of a
Brownian motion onto a closed Weyl chamber is another Brownian motion normally
reflected on the walls of the chamber. Our proof is probabilistic and the
decomposition we obtain may be seen as a multidimensional extension of Tanaka's
formula for linear Brownian motion. The paper is closed with a description of
the boundary process through the local times at zero of the distances from the
initial process to the facets.
