On Convergence to Equilibrium Distribution, I. The Klein - Gordon Equation with Mixing

Abstract

Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant or variable coefficients. We study the distribution $\mu_t$ of the random solution at time $t\in\R$. We assume that the initial probability measure $\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that $\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using an `averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.