Infinite Dimensional Stochastic Differential Equations for Dyson's Model

TL;DR

This paper establishes the existence, uniqueness, and convergence of solutions for an infinite-dimensional SDE modeling Dyson's Brownian Motion, extending the understanding of such processes for all b1.

Contribution

It provides a rigorous construction and analysis of an infinite-dimensional SDE for Dyson's model applicable to various initial conditions, including the sine process.

Findings

Proves strong existence and pathwise uniqueness of the SDE for all b1.

Shows convergence from finite to infinite-dimensional SDEs.

Confirms the determinantal formula for =2 case.

Abstract

In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional Stochastic Differential Equation (SDE) corresponding to the bulk limit of Dyson's Brownian Motion (DBM), for all $\beta\geq 1$. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration $\{x_i\}=\mathbb{Z}$ and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (2010) for the solution of this SDE at $\beta=2$.