ON CONVERGENCE TO THE EQUILIBRIUM DISTRIBUTION: HARMONIC CRYSTAL WITH MIXING

Consider the oscillations of the harmonic crystal in ℝⁿ with an arbitrary n ≥ 1. We study the distribution μ_t of the random solution at time t ∈ ℝ. The initial measure μ₀ has translation-invariant correlation matrices, zero mean and a finite mean energy density. It also satisfies Rosenblatt- or Ibragimov-Linnik type mixing condition. The main result is the convergence of μ_t to a Gaussian measure as t → ∞. The proof is based on the long time asymptotics of the Green function and on the S.N.Bernstein's "room-corridors" method.