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We consider coupled fast-slow stochastic processes, where the averaged slowmotion is given by a two-dimensional Hamiltonian system with multiple criticalpoints. On a proper time scale, the evolution of the first integral convergesto a diffusion process on the corresponding Reeb graph, with certain gluingconditions specified at the interior vertices. This limiting process is similarto that in the case of additive white noise perturbations of Hamiltoniansystems considered by M. Freidlin and A. Wentzell, but now the analysis of thebehavior near the critical points requires new interesting techniques.

Fast oscillating random perturbations of Hamiltonian systems