Infinite Chain of Harmonic Oscillators Under the Action of the Stationary Stochastic Force

Abstract

We consider countable system of harmonic oscillators on the real line with quadratic interaction potential with finite support and local external force (stationary stochastic process) acting only on one fixed particle. In the case of positive definite potential and initial conditions lying in $l_2(\mathbb{Z})$-space the perpesentation of the deviations of the particles from their equilibrium points are found. Precisely, deviation of each particle could be represented as the sum of some stationary process (it is also time limiting process in distribution for that function) and the process which converges to zero as $t\rightarrow+\infty$ with probability one. The time limit for the mean energy of the whole system is found as well.