Non-linear hydrodynamic fluctuations around equilibrium

We formulate a scheme describing the fluctuations in a system obeying the non-linear hydrodynamic equations. The random fluxes are assumed to be Gaussian processes with white noise. It is shown that the usual expressions for the systematic parts of the dissipative fluxes are consistent with this assumption, provided that the Onsager coefficients are constants. The linear response of the system to a small external force field is studied and the relevant fluctuation- dissipation theorems are derived.