Some martingales associated with multivariate Bessel processes

TL;DR

This paper introduces new martingales for multivariate Bessel processes on Weyl chambers, linking them to symmetric functions and orthogonal polynomials, with implications for random matrix theory.

Contribution

It presents novel space-time-harmonic functions and martingales for Bessel processes, independent of a parameter, and connects these to characteristic polynomials in random matrix theory.

Findings

Derived space-time-harmonic functions for Bessel processes

Expressed expectations of characteristic polynomials via orthogonal polynomials

Established connections to random matrix theory interpretations

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