  We study the energy diffusion in a chain of anharmonic oscillators where the
Hamiltonian dynamics is perturbed by a local energy conserving noise. We prove
that under diffusive rescaling of space-time, energy fluctuations diffuse and
evolve following an infinite dimensional linear stochastic differential
equation driven by the linearized heat equation. We also give variational
expressions for the thermal diffusivity and some upper and lower bounds.
