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 $\mathbb{Z}^d$ of $\mathbb{R}^d$. The initial date is assumed to be a random function that is close to two spatially homogeneous (with respect to the subgroup $\mathbb{Z}^d$) processes when $\pm x_1>a$ with some $a>0$. We study the distribution $\mu_t$ of the solution at time $t\in\mathbb{R}$ and prove the weak convergence of $\mu_t$ to a Gaussian measure $\mu_\infty$ as $t\to\infty$. Moreover, we prove the convergence of the correlation functions to a limit and derive the explicit formulas for the covariance of the limit measure $\mu_\infty$. We give an application to Gibbs measures.