Global fluctuations for 1D log-gas dynamics. (2) Covariance kernel and support

TL;DR

This paper studies the hydrodynamic limit and fluctuations of a generalized Dyson's Brownian motion log-gas system, providing explicit formulas for the covariance kernel and analyzing the evolution of the system's support.

Contribution

It offers new regularity results, support evolution analysis, and explicit covariance kernel formulas for the fluctuation process in the 1D log-gas dynamics.

Findings

Explicit covariance kernel for the fluctuation process.

Regularity results for the McKean-Vlasov equation.

Analysis of the support evolution over time.

Abstract

We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ d\lambda_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(\lambda_t^i) dt+ \frac{\beta}{2N} \sum_{j\not=i} \frac{dt}{\lambda^i_t-\lambda^j_t}, \qquad i=1,\ldots,N, $ with $\beta>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown by the author to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation. We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho_t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.