On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing

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, transla...

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Bibliographic Details
Published in:arXiv.org 2005-08
Main Authors: Dudnikova, T V, Komech, A I, Ratanov, N E, Suhov, Yu M
Format: Article
Language:English
Subjects:
Asymptotic properties
Coefficients
Convergence
Covariance matrix
Flux density
Wave equations
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Summary: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.
ISSN:2331-8422