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-timeharmonic functions and thus martingales for these processes $$(X_t)_{t\ge0}$$ which are independent from one parameter of these processes. As a consequence, $$p_t(y):= \mathbb{E}(\prod_{i=1}^N (y-X_t^i))$$ 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.