On random perturbations of Hamiltonian systems with many degrees of freedom

We consider a class of random perturbations of Hamiltonian systems with many degrees of freedom. We assume that the perturbations consist of two components: a larger one which preserves the energy and destroys all other first integrals, and a smaller one which is a non-degenerate white noise type process. Under these assumptions, we show that the long time behavior of such a perturbed system is described by a diffusion process on a graph corresponding to the Hamiltonian of the system. The graph is homeomorphic to the set of all connected components of the level sets of the Hamiltonian. We calculate the differential operators which govern the process inside the edges of the graph and the gluing conditions at the vertices.