On Convergence to Equilibrium Distribution,I.¶The Klein–Gordon Equation with Mixing |
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| Authors: | T V Dudnikova A I Komech E A Kopylova Suhov |
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| Institution: | 1.Mathematics Department, Elektrostal Polytechnical Institute, Elektrostal, 144000 Russia.?E-mail: misis@elsite.ru,RU;2.Mechanics and Mathematics Department, Moscow State University, Moscow, 119899 Russia.?E-mail:komech@mech.math.msu.ru,RU;3.Physics and Applied Mathematics Department, Vladimir State University, Vladimir, Russia.?E-mail: ks@vpti.vladimir.ru,RU;4.Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of?Cambridge, Cambridge, UK. E-mail: Y.M.Suhov@statslab.cam.ac.uk},GB |
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| 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 μ0 has zero mean, a translation-invariant covariance, and a finite mean energy density. We also assume that μ0 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.
Received: 4 January 2001 / Accepted: 2 July 2001 |
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