2005/08/19 by T. V. Dudnikova, A. I. Komech, N. E. Ratanov +1
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.PR #msc:35L05 #msc:60F05
published as Journal of Statistical Physics 108 (2002), no.4, 1219-1253 · 27 pages
arxiv created 2005/08/19 · arxiv updated 2009/12/01
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.