Convergence to stationary states for infinite harmonic systems

We study the evolution of the states for one-dimensional infinite harmonic systems, interacting through a translation invariant force of rapid decrease. We prove that for a large class of initial states convergence to a Gaussian limiting state, as time goes to infinity, is equivalent to convergence of the covariance. The main assumption on the initial states is a kind of weak dependence between distant regions (mixing condition). We prove also...