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MiS Preprint
25/2000
On convergence to statistic equilibrium in wave equations with mixing
Tatiana V. Dudnikova, Alexander Komech and Nikita E. Ratanov
Abstract
The wave equation with constant or variable coefficients in the whole space $\mathbb{R}^n$ with an arbitrary odd $n \geq 3$ is considered. The initial datum is a translation-invariant random function with zero expectation and finite mean density of the energy, which also fits the mixing condition of Ibragimov-Linnik-Rosenblatt type. We study the distribution $\mu_t$ of the random solution at the moment $t \in \mathbb{R}$. The main result is the convergence of $\mu_t$ to some Gaussian measure as $t \to \infty$. This is the central limit theorem for linear wave equations. For the case of constant coefficients the proof is based on a new analysis of Kirchhoff's and Herglotz-Petrovskii's integral representations of the solution and on S.N.Bernstein's "room-corridors" method. The case of variable coefficients is reduced to constant coefficients. For this purpose the scattering theory for infinite energy solutions is constructed. The relation to Gibbs measures is discussed. The investigation is inspired by the problems of the mathematical foundation of the statistical physics.
