### Abstract

Consider the Klein-Gordon equation (KGE) in ℝ^{n} , n ≥ 2, with constant or variable coefficients. We study the distribution μ _{t} of the random solution at time t ∈ ℝ. We assume that the initial probability measure μ_{0} has zero mean, a translation-invariant covariance, and a finite mean energy density. We also assume that μ_{0} satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of μ _{t} to a Gaussian probability measure as t→∞ which gives a Central Limit Theorem for the KGE. 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 an "averaged" version ofthe 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) | 1-32 |

Number of pages | 32 |

Journal | Communications In Mathematical Physics |

Volume | 225 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*225*(1), 1-32. https://doi.org/10.1007/s002201000581