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 μt 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 μt 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.