We describe the long-time behavior of the nonlinear pendulum perturbed by a small noise. To derive this asymptotics, one has to consider diffusion processes on the graph corresponding to the Hamiltonian of the pendulum.
"A remark on random perturbations of the nonlinear pendulum." Ann. Appl. Probab. 9 (3) 611 - 628, August 1999. https://doi.org/10.1214/aoap/1029962806