Infinite-dimensional stochastic differential equations related to random matrices

TL;DR

This paper solves infinite-dimensional stochastic differential equations for particles interacting via 2D Coulomb potentials, linking them to random matrix theory and establishing new integration by parts formulas for related measures.

Contribution

It introduces methods to solve ISDEs with long-range Coulomb interactions and connects these solutions to the Ginibre and Dyson measures in random matrix theory.

Findings

Established an integration by parts formula for Ginibre and Dyson measures.

Proved that particles in the Ginibre ensemble satisfy multiple ISDEs.

Highlighted the distinct properties of Coulomb-interacting particles compared to Ruelle's class interactions.

Abstract

We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle's class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs.