Abstract
We consider wave and Klein-Gordon equations in the whole space ℝn with arbitraryn≥2. We assume initial data to be homogeneous random functions in ℝn with zero expectation and finite mean density of energy. Moreover, we assume initial data fit mixing condition of Ibragimov-Linnik type. We consider the distributions of the random solution at the moment of timet. The main results mean the convergence of this distribution to some Gaussian measure ast→∞. This is a central limit theorem for wave and Klein-Gordon equations. The limit Gaussian measures are invariant measures for equations considered. Corresponding stationary random solutions are ergodic and mixing in time. The results are inspired by mathematical problems of statistical physics.
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Conferenza tenuta il 27 marzo 1995
Supported partly by French-Russian A.M. Liapunov Center of Moscow State University and by research grants of AMS, ISF (ND 7000), ISF and Russian Government (ND 7300), RFFI (no. 93-011-16035), University of Russia.
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Komech, A.I. Stabilization of statistics in wave and Klein-Gordon equations with mixing. Scattering theory for infinite energy solutions. Seminario Mat. e. Fis. di Milano 65, 9–22 (1995). https://doi.org/10.1007/BF02925249
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DOI: https://doi.org/10.1007/BF02925249