  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.
