Asymptotic behaviour of the statistical solutions of linear differential equations

We study the convergence of the statistical solution of the wave equation. More precisely we consider the Cauchy problem with random initial data. At first we suppose that initial data are homogeneous random field with mixing (in the sense of M. Rosenblatt). We prove weak convergence of the distributions of the solutions to the Gaussian measure. Also we consider the Cauchy problem for the wave equation with random coefficient. An asymptotical behaviour of solutions is discussed, as t --> infinity. The answer depends on the initial date. In the cases of summable and periodic initial data the asymptotics is precisely described. The limiting distribution of the solution is non-Gaussian.