We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, $ dlambda_t^i=frac{1}{sqrt{N}} dW_t^i - V(lambda_t^i) dt+ frac{beta}{2N} sum_{j ot=i} frac{dt}{lambda^i_t-lambda^j_t}, qquad i=1,ldots,N, $ with $beta>1$, sometimes called generalized Dysons 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 $Ntoinfty$ 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.
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