Diffusion and superdiffusion of energy in one dimensional systems of oscillators

We consider a system of harmonic oscillators with stochastic perturbations of the dynamics that conserve energy and momentum. In the one dimensional unpinned case, under proper space-time rescaling, Wigner distribution of energy converges to the solution of a fractional heat equation (with power 3/4 for the laplacian). For pinned systems or in dimension 3 or higher, we prove normal diffusive behaviour. Similar results are also obtained for space-time energy correlations in equilibrium. This is in agreement with previous 'weak noise' limits, passing through a kinetic equation, and conjectured behaviour for beta-FPU chains (quartic symmetric interaction). Joint works with Tomasz Komorowski and Giada Basile.