Cores of Dirichlet forms related to random matrix theory

TL;DR

This paper establishes that polynomial functions form cores for Dirichlet forms associated with infinite-dimensional interacting Brownian motions, such as Dyson's and Airy processes, which are linked to random matrix theory.

Contribution

It proves that polynomial functions are cores for Dirichlet forms of certain infinite-dimensional stochastic processes related to random matrices.

Findings

Polynomial functions form cores for Dirichlet forms of these processes.

Results will facilitate proving the equivalence of different constructions of these stochastic dynamics.

Applications to random matrix theory models like Dyson and Airy Brownian motions.

Abstract

We prove the sets of polynomials on configuration spaces are cores of Dirichlet forms describing interacting Brownian motion in infinite dimensions. Typical examples of these stochastic dynamics are Dyson's Brownian motion and Airy interacting Brownian motion. Both particle systems have logarithmic interaction potentials, and naturally arise from random matrix theory. The results of the present paper will be used in a forth coming paper to prove the identity of the infinite-dimensional stochastic dynamics related to the random matrix theories constructed by apparently different methods: the method of space-time correlation functions and that of stochastic analysis.