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

TL;DR

This paper constructs and analyzes infinite-dimensional stochastic differential equations (ISDEs) associated with Airy random point fields, establishing existence, uniqueness, and connection to finite particle systems, advancing the understanding of stochastic dynamics in random matrix theory.

Contribution

The paper introduces a new method to prove existence and uniqueness of strong solutions for ISDEs related to Airy point fields, linking finite particle dynamics to infinite-dimensional limits.

Findings

Constructed reversible diffusion processes for Airy random point fields.

Proved existence and pathwise uniqueness of strong solutions to the ISDEs.

Connected solutions to finite N-particle systems in the soft-edge limit.

Abstract

The Airy$_{\beta }$ random point fields ($ \beta = 1,2,4$) are random point fields emerging as the soft-edge scaling limits of eigenvalues of Gaussian random matrices. We construct the unlabeled diffusion reversible with respect to the Airy$_{\beta }$ random point field for each $ \beta = 1,2,4$. We identify the infinite-dimensional stochastic differential equations (ISDEs) describing the labeled stochastic dynamics for the unlabeled diffusion mentioned above. We prove the existence and pathwise uniqueness of strong solutions of these ISDEs. Furthermore, the solution of the ISDE is the limit of the solutions of the stochastic differential equations describing the dynamics of the $ N $-particle system in the soft-edge limit. We thus establish the construction of the stochastic dynamics whose unlabeled dynamics are reversible with respect to the Airy random point fields. When $ \beta=2 $, the solution equals the stochastic dynamics defined by the space-time correlation functions obtained by Pr\"ahofer--Spohn, Johansson, Katori--Tanemura, and Corwin--Hammond, among others. We develop a new method whereby these ISDEs have unique, strong solutions. We expect that our approach is valid for other soft-edge scaling limits of stochastic dynamics arising from the random matrix theory.