Linear Hamiltonian systems under microscopic random influence

It is known that a linear Hamiltonian system has too many invariant measures; thus the problem of convergence to Gibbs measure makes no sense. We consider linear Hamiltonian systems of arbitrary finite dimension and prove that, under the condition that one distinguished coordinate is subjected to dissipation and white noise, for “almost any” Hamiltonians and “almost any” initial conditions, there exists a unique limiting distribution. Moreover, this distribution is Gibbsian with the temperature depending on the dissipation and on the variance of the white noise.