Weakly harmonic oscillators perturbed by a conservative noise

We consider a chain of weakly harmonic coupled oscillators perturbed by a conservative noise. We show that by tuning accordingly the coupling constant, energy can diffuse like a Brownian motion or superdiffuse like a maximally 3/2-stable asymmetric Lévy process. For a critical value of the coupling, the energy diffusion is described by a family of Lévy processes which interpolate between these two processes.