Stabilization of the statistical solutions for large times for a harmonic lattice coupled to a Klein–Gordon field.

We consider the Cauchy problem for the Hamiltonian system consisting of the Klein–Gordon field and an infinite harmonic crystal. The dynamics of the coupled system is translation-invariant with respect to the discrete subgroup of . The initial date is assumed to be a random function that is close to two spatially homogeneous (with respect to the subgroup ) processes when with some . We study the distribution of the solution at time and prove the weak convergence of to a Gaussian measure as . Moreover, we prove the convergence of the correlation functions to a limit and derive the explicit formulas for the covariance of the limit measure . We give an application to Gibbs measures. [ABSTRACT FROM AUTHOR]