Physics – Mathematical Physics
Scientific paper
2005-08-19
Journal of Statistical Physics 108 (2002), no.4, 1219-1253
Physics
Mathematical Physics
27 pages
Scientific paper
The paper considers the wave equation, with constant or variable coefficients in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the distribution $\mu_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It is assumed that the initial measure $\mu_0$ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that $\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian measure $\mu_\infty$ as $t\to\infty$, 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.
Dudnikova T. V.
Komech A. I.
Ratanov N. E.
Suhov Yu. M.
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