Reflected Brownian motion in Weyl chambers

We supply two different descriptions of the pushing process driving the reflected Brownian motion in Weyl chambers, when the latter domains are simplexes. The first one shows that a simple root lies in one and only one orbit if and only if the pushing process in the direction of that simple root increases as the sum of all the Brownian local times in the directions of the orbit's positive elements. The last one shows that the pushing process may be written as the sum of an inward normal vector at the chamber's boundary and an inward normal vector at the origin, yielding a kind of a multivoque stochastic differential equation for the reflected process. We finally give a particles system interpretation of the reflected process and construct a multidimensional skew Brownian motion.