On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing |
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| Authors: | T V Dudnikova A I Komech N E Ratanov Y M Suhov |
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| Institution: | (1) Mathematics Department, Elektrostal Polytechnical Institute, Elektrostal, 144000, Russia;(2) Mechanics and Mathematics Department, Moscow State University, Moscow, 119899, Russia;(3) Economics Department, Chelyabinsk State University, Chelyabinsk, 454021, Russia;(4) Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, CB3 0WB, United Kingdom;(5) Institüt für Mathematik, Universität Wien, Vienna, 1090, Austria |
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| Abstract: | The paper considers the wave equation, with constant or variable coefficients in 
n
, with odd n 3. We study the asymptotics of the distribution 
t
of the random solution at time t  as t   . It is assumed that the initial measure
0 has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that
0 satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of
t
to a Gaussian measure
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. |
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| Keywords: | Wave equation Cauchy problem random initial data mixing condition Fourier transform convergence to a Gaussian measure covariance functions and matrices |
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