Some martingales associated with multivariate Jacobi processes and Aomoto’s Selberg integral

Abstract

We study β-Jacobi diffusion processes on alcoves in RN, depending on 3 parameters. Using elementary symmetric functions, we present space–time-harmonic functions and martingales for these processes (Xt)t≥0 which are independent from one parameter. This leads to a formula for E(∏i=1N(y−Xt,i)) in terms of classical Jacobi polynomials. For t→∞ this yields a corresponding formula for Jacobi ensembles and thus Aomoto’s Selberg integral.