Dyson Brownian motion on a Jordan curve

TL;DR

This paper rigorously constructs Dyson Brownian motion confined to a Jordan curve, analyzing its properties, convergence to equilibrium, large deviations, and mean-field limits under smoothness assumptions.

Contribution

It provides a rigorous framework for Dyson Brownian motion on Jordan curves and derives key equations and properties in this setting.

Findings

Convergence to stationary Coulomb gas distribution

Derivation of Fokker-Planck-Kolmogorov equation

Analysis of large deviations and mean-field limit

Abstract

Zabrodin recently proposed a generalization of Dyson Brownian motion to a setting where the particles are confined to a smooth Jordan curve in the plane. In this paper, we discuss a rigorous construction of such a process on a rectifiable Jordan curve and study some of its basic properties. Under further smoothness assumptions, we derive the associated Fokker-Planck-Kolmogorov equation, prove convergence towards the stationary Coulomb gas distribution, study large deviations at low temperature, and derive the limiting mean-field McKean--Vlasov equation in the many-particle limit.