On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

Abstract

The paper considers the wave equation, with constant or variable coefficients in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the distribution $μ_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It is assumed that the initial measure $μ_0$ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that $μ_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of $μ_t$ to a Gaussian measure $μ_\infty$ as $t\to\infty$, which gives a Central Limit Theorem (CLT) for the wave equation. 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 a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.