On the asymptotical normality of statistical solutions for wave equations coupled to a particle

Abstract

We consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein-Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution µ t of the random solution at time moments t ∈ R. The main result is the convergence of µ t to a Gaussian probability measure as t → ∞. The mixing properties of the limit measures are studied. The application to the case of Gibbs initial measures is given.