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. 1. overview By ‘the reflected Brownian motion’ (shorthand RBM), it is often meant the absolute value of a real Brownian motion. This process has gained much fame since it is not an Itô’s semimartingale and due to the celebrated Lévy’s representation in relation with Skorohod’s problem ([10]). More precisely, if B is a standard real Brownian motion, then there exist a standard real Brownian motion β and an increasing process L0(B) such that (1) |Bt| = βt + Lt (B). The process L0 is known as the local time at 0 of B since the following representation holds (Ch.VI in [10]) Lt (B) = lim ǫ→0 1 2ǫ ∫ t 0 1{|Bs|≤ǫ}ds = 1 2 Lt (|B|), and it gives the explicit Doob-Meyer’s decomposition of the submartingale |B|. It also shows that L0(B) is adapted with respect to the natural filtration F |B| of |B|. Besides, the support of dL0(B) is contained in the set {t, Bt = 0} and the identity in law of trajectories due to Paul Lévy holds ([10]) (|B|, L(B)) d = (S −B,S), St := sup 0≤s≤t Bs.