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

TL;DR

This paper develops martingales for multivariate Jacobi processes using elementary symmetric functions, leading to new formulas for expectations related to Jacobi ensembles and Aomoto's Selberg integral.

Contribution

It introduces space-time-harmonic functions and martingales for beta-Jacobi processes that are independent of one parameter, connecting stochastic processes with classical integrals.

Findings

Derived formulas for expectations involving Jacobi polynomials.

Established connections between Jacobi processes and Aomoto's Selberg integral.

Provided explicit martingales for multivariate Jacobi diffusions.

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 $(X_t)_{t\ge0}$ 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.