We consider a purely harmonic chain of oscillators perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: volume and energy. At the hydrodynamic level, in the diffusive time scale, we show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or according to a non-linear system of coupled parabolic equations. At the level of fluctuations, we show that, also in diffusive time scale, under any initial measure, the volume fluctuation field converges. The proofs are based on a precise analysis of the two-point correlation function and a uniform fourth moment bound. We also discuss some open problems and the technical issues faced when studying higher order correlation functions in this models of stochastic oscillators. Joint work with Patricia Gonçalves (IST Lisbon) and Kohei Hayashi (University of Osaka).