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 $(X_t)_{ t\ge0}$ 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.
Categories: math.PR math-ph math.CA math.MP math.RT