Stabilization of statistical solutions to the wave equation in the even-dimensional space

Consider the wave equations in $\mathbb R^n$, with constant or variable coefficients for even $n\ge 4$. The initial datum is a random function with a finite mean density of energy that satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. It is assumed that the initial random function converges to two distinct space-homogeneous processes as $x_n\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution $\mu_t$ of the random solution at a time $t\in\mathbb R$. The main result is the convergence of $\mu_t$ to a Gaussian measure as $t\to\infty$.