### Abstract

The paper considers the wave equation, with constant or variable coefficients in R^{n}, with odd n ≥ 3. We study the asymptotics of the distribution μ_{1} of the random solution at time t ∈ R as t ∞. It is assumed that the initial measure μ_{0} has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that μ_{0} satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of μ_{1} 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.

Original language | English (US) |
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Pages (from-to) | 1219-1253 |

Number of pages | 35 |

Journal | Journal of Statistical Physics |

Volume | 108 |

Issue number | 5-6 |

DOIs | |

State | Published - Dec 1 2002 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*108*(5-6), 1219-1253. https://doi.org/10.1023/A:1019755917873