We consider random perturbations of Hamiltonian systems. Under suitable conditions the long time behavior of such a perturbed system is described by a diffusion process on a graph related to the Hamiltonian of the system.
We present results for non-linear oscillators with one degree of freedom, especially for the non-linear pendulum perturbed by white noise, and results for dynamical systems with many degrees of freedom.
We calculate the characteristics of the diffusion process on the graph and present some applications.
All results are joint work with Mark Freidlin from the University of Maryland, U.S.A.