Some martingales associated with multivariate Bessel processes

Abstract

We study Bessel processes on Weyl chambers of types A and B on ${\mathbb{R}}^N$. Using elementary symmetric functions, we present several space-time-harmonic functions and thus martingales for these processes $(X_t{)}_{t\ge 0}$ which are independent from one parameter of these processes. As a consequence, $p(y):=\mathbb{E}({\prod }_{i=1}^N(y-X_t^i))$ can be expressed via classical orthogonal polynomials. Such formulas on characteristic polynomials admit interpretations in random matrix theory where they are partially known by Diaconis, Forrester, and Gamburd.