Cores of Dirichlet forms related to random matrix theory

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.Comment: 6 pages, revised version, published in PJA in 201

Infinite-dimensional stochastic differential equations arising from Airy random point fields

We identify infinite-dimensional stochastic differential equations (ISDEs) describing the stochastic dynamics related to Airy$_{\beta }$ random point fields with $\beta =1,2,4$. We prove the existence of unique strong solutions of these ISDEs. When $\beta = 2$, this solution is equal to the stochastic dynamics defined by the space-time correlation functions obtained by Spohn and Johansson among others. We develop a new method to construct a unique, strong solution of ISDEs. We expect that our approach is valid for other soft-edge scaling limits of stochastic dynamics arising from the random matrix theory.Comment: 55 page

Stochastic differential equations for infinite particle systems of jump type with long range interactions

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a L\'evy process, and the interaction between particles is determined by the long-range interaction potential. The potential is of Ruelle's class or logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.Comment: 60page