β-Jacobi processes

Abstract

We define and study a [0, 1]^m-valued process depending on three positive real parameters p, q, β that specializes for β = 1, 2 to the eigenvalues process of the real and complex matrix Jacobi processes on the one hand and that has the distribution of the β-Jacobi ensemble as stationary distribution on the other hand. We first prove that this process, called β-Jacobi process, is the unique strong solution of the stochastic differential equation defining it provided that β > 0, p ∧ q > m – 1 + 1/β. When specialized to β = 1, 2, our results actually improve well known results on eigenvalues of matrix Jacobi processes. While proving the strong uniqueness, the generator of the β-Jacobi process is mapped into the radial part of the Dunkl–Cherednik Laplacian associated with the non reduced root system of type BC. The transformed process is then valued in the principal Weyl alcove and this allows to define the Brownian motion in the Weyl alcove corresponding to all multiplicities equal one. Second, we determine, using stochastic calculus and a comparison theorem, the range of β, p, q for which the m components of the β-Jacobi process first collide, the smallest one reaches 0 and the largest one reaches 1. This is equivalent to the first hitting time of the boundary of the principal Weyl alcove by the transformed process. Finally, we write down its semi group density.