Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle

The Hamiltonian system formed by a Klein-Gordon vector field and a particle in ℝ³ is considered. The initial data of the system are given by a random function, with finite mean energy density, which also satisfies a Rosenblatt- or Ibragimov-type mixing condition. Moreover, initial correlation functions are assumed to be translation invariant. The distribution μ _t of the solution at time t ∈ ℝ is studied. The main result is the convergence of μ _t to a Gaussian measure as t → ∞, where μ_∞ is translation invariant.     

Behavior for Large Time of a Two-Component Chain of Harmonic Oscillators

We consider a model consisting of two semi-infinite chains of harmonic oscillators with a special particle subject to a harmonic external potential.We study the Cauchy problem for this model. The main goal is to derive the dispersive bounds for the solutions in the energy weighted norms.