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We consider the hydrodynamic limit in the macroscopic regime of the coupledsystem 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 thedissipative dynamics of a log-gas of $N$ equal charges with equilibrium measurecorresponding to a $\beta$-ensemble, with sufficiently regular convex potential$V$. The limit $N\to\infty$ is known to satisfy a mean-field Mc Kean-Vlasovequation. Fluctuations around this limit have been shown by the author todefine a Gaussian process solving some explicit martingale problem written interms of a generalized transport equation.  We prove a series of results concerning either the Mc Kean-Vlasov equationfor the density $\rho_t$, notably regularity results and time-evolution of thesupport, or the associated hydrodynamic fluctuation process, whose space-timecovariance kernel we compute explicitly.

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