2005/08/22 by T. V. Dudnikova, A. I. Komech, H. Spohn
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.PR #msc:60Fxx #msc:60Gxx #msc:82-xx
published as Markov Processes and Related Fields 8 (2002), no.1, 43-80 · 30 pages
arxiv created 2005/08/22 · arxiv updated 2009/12/01
Consider the wave equation with constant or variable coefficients in \R3. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The random function converges to different space-homogeneous processes as x3→±∞, with the distributions μ_±. We study the distribution μt of the random solution at a time t∈\R. The main result is the convergence of μt to a Gaussian translation-invariant measure as t→∞ that means central limit theorem for the wave equation. The proof is based on the Bernstein `room-corridor' argument. The application to the case of the Gibbs measures μ_±=g_± with two different temperatures T± is given. Limiting mean energy current density formally is -∞⋅ (0,0,T+ -T-) for the Gibbs measures, and it is finite and equals to -C(0,0,T+ -T-) with C>0 for the convolution with a nontrivial test function.