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

We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations,\[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \]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 [39] 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.