Global fluctuations for 1D log-gas dynamics

We study in this article the hydrodynamic limit in themacroscopic regime of the coupled system of stochastic differential equations,

(0.1)$d{\lambda }_t^i=\frac{1}{\sqrt{N}}d{W}_t^i?V^{\prime }\left({\lambda }_t^i\right)dt+\frac{\beta }{2N}\sum _{j\ne i}\frac{dt}{{\lambda }_t^i?{\lambda }_t^j},i=1,\dots ,N,$

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. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE.The proof is very much indebted to the harmonic potential case treated in Israelsson (2001). Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process.