Orthogonal Intertwiners for Infinite Particle Systems in The Continuum

Abstract

This article focuses on a system of sticky Brownian motions, also known as Howitt-Warren martingale problem, and correlated Brownian motions and shows that infinite-dimensional orthogonal polynomials intertwine the dynamics of infinitely many particles and their $n$-particle evolution. The proof is based on two assumptions about the model: information about the reversible measures for the $n$-particle dynamics and consistency. Additionally, explicit formulas for the polynomials are used, including a new explicit formula for infinite-dimensional Meixner polynomials, the orthogonal polynomials with respect to the Pascal process. As an application of the intertwining relations, new reversible measures for the infinite-particle dynamics are obtained.