Energy Diffusion in Harmonic System with Conservative Noise

Lattice networks of oscillators have been considered for a long time as good models for studying macroscopic energy trasport and its diffusion, i.e. for obtaining, on a macroscopic space-time scale, heat equation and Fourier law of conduction ([1]). It is well understood that the diffusive behavior of the energy is due to the non-linearity of the interactions, and that purely deterministic harmonic systems have a ballistic transport of energy (cf. [14]). On the other hand, non-linear dynamics are very difficult to study and even the convergence of the Green-Kubo formula defining the macroscopic thermal conductivity is an open problem. In fact in some cases, like in one dimensional unpinned systems, it is expected (and observed numerically) an infinite conductivity and a superdiffusion of the energy (cf. [12, 11]). In order to model the phonon scattering due to the dynamics, various stochastic perturbation of the harmonic dynamics have been proposed, where the added random dynamics conserves the energy. In [7] Fourier Law is proven for an harmonic chain with stocastic dynamics that conserves only energy. In [2, 3] is studied the Green-Kubo formula for a stochastic perturbation that conserves energy and momentum. It is proven there that conductivity is finite for pinned systems, or unpinned if dimension is greater than 2. Qualitatively this agrees to what it is expected for deterministic anharmonic dynamics. In this article we consider the same stochastic dynamics as in [2, 3] and we prove that in the cases when conductivity is finite, the energy