Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit

We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling limit. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random terms conservative of such quantities. We prove that these fluctuations evolve macroscopically following the linearized Euler equations with the corresponding boundary conditions. Furthermore, we prove that such linearized evolution holds in some time scales larger than the hyperbolic one.