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.