Hermite trace polynomials and chaos decompositions for the Hermitian Brownian motion

TL;DR

This paper introduces Hermite trace polynomials related to Hermitian Brownian motion, exploring their combinatorial properties, orthogonality, and chaos decompositions, with connections to matrix Hermite polynomials.

Contribution

It defines Hermite trace polynomials for a parameter q, establishes their combinatorial and orthogonal properties, and develops chaos decompositions for stochastic integrals related to Hermitian Brownian motion.

Findings

Hermite trace polynomials form an orthogonal basis for certain q values.

The state induced by these polynomials matches the expectation of Hermitian Brownian motion.

Chaos decompositions are established for stochastic integrals involving these polynomials.

Abstract

For a non-zero parameter $q$, we define Hermite trace polynomials, which are multivariate polynomials indexed by permutations. We prove several combinatorial properties for them, such as expansions and product formulas. The linear functional determined by these trace polynomials is a state for $q = \frac{1}{N}$ for $N$ a non-zero integer. For such $q$, Hermite trace polynomials of different degrees are orthogonal. The product formulas extend to the closure with respect to the state. The state can be identified with the expectation induced by the $N \times N$ Hermitian Brownian motion. Hermite trace polynomials are martingales for this Brownian motion, while the elements in the closure can be interpreted as stochastic integrals with respect to it. Using the grading on the algebra, we prove several chaos decompositions for such integrals, as well as analyze corresponding creation and annihilation operators. In the univariate, pure trace polynomial case, trace Hermite polynomials can be identified with the Hermite polynomials of matrix argument.