On convergence to equilibrium distribution, II. The wave equation in odd dimensions, with mixing (2002)
The paper considers the wave equation, with constant or variable coefficients in ?n, with odd n?3. We study the asymptotics of the distribution ? t of the random solution at time t ? ? as t ? ?. It is assumed that the initial measure ? 0 has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that ? 0 satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of ? t to a Gaussian measure ? ? 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.