On convergence to equilibrium distribution, II. The wave equation in odd dimensions, with mixing

The paper considers the wave equation, with constant or variable coefficients in Rⁿ, with odd n ≥ 3. We study the asymptotics of the distribution μ₁ of the random solution at time t ∈ R as t ∞. It is assumed that the initial measure μ₀ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that μ₀ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of μ₁ to a Gaussian measure μ_∞ as t → ∞, 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.