Large system of oscillators with ultralocal stochastic stationary external field influence

In this paper influence of external force, assumed to be random stationary process, on the behavior of large Hamiltonian particle systems is studied. The Hamiltonian system is assumed to have quadratic interaction, and the external influence is assumed to be local. More exactly, the external force acts on only one fixed particle. Such systems were studed earlier, it is given short review of the previous papers. In our case, when the external force is a stationary random process in the wider sense, large time asymptotics of the mean energy of the system is studied. Main result is the characterization of 4 different cases for the spectrum of the matrix of quadratic interaction and the spectral density of the correlation function of the stationary random process, which give different asymptotic behaviour of the trajectories and of the mean energy. Typical behaviour appears to be either uniform boundedness or quadratic growth of the mean energies.