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

Abstract

We study $\beta$-Jacobi diffusion processes on alcoves in ${\mathbb{R}}^N$, depending on 3 parameters. Using elementary symmetric functions, we present space-time-harmonic functions and martingales for these processes $(X_t{)}_{t\ge 0}$ which are independent from one parameter. This leads to a formula for $\mathbb{E}({\prod }_{i=1}^N(y-X_{t,i}))$ in terms of classical Jacobi polynomials. For $t\to \infty$ this yields a corresponding formula for Jacobi ensembles and thus Aomoto's Selberg integral.