On convergence to equilibrium for wave equations in $\mathbb R^n$, with odd $n\ge3$

T. V. Dudnikova


Abstract: Consider the wave equations in $\mathbb R^n$, with $n\ge3$ and odd, with constant or variable coefficients. 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 different 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 $t\to\infty$ that means a central limit theorem for the wave equations. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_\pm$ is given.