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

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