We develop rigorous methods to study one to one and one to two resonance problems as a random perturbation of four-dimensional Hamiltonian systems. The focus of this paper is the development of general techniques of stochastic averaging of randomly-perturbed four-dimensional integrable Hamiltonian systems with an elliptic fixed point, and certain nontrivial (yet generic) resonance condition. Classical stochastic averaging makes use of these integrable structures to identify a reduced diffusive model on a space which encodes the structure of the fixed points.

The interest of this paper is when classical methods fail because the original Hamiltonian system has resonances, which give rise to singularities in the lower-dimensional description}. At these singularities, glueing conditions will be derived, these glueing conditions completing the specification of the dynamics of the reduced model. The general dimensional reduction techniques developed here, consists of a sequence of averaging procedures that are uniquely adapted to study noisy mechanical systems.

Using the above theory, we develop numerical algorithms to computationally study the reduced stochastic models for mechanical systems. Whereas the original system is 4-dimensional, the state space of the reduced model is a 2-dimensional graph; thus we are interested in various relevant partial differential equations on this graph (viz. the stationary and time-dependent Fokker-Planck equations for transition densities and stationary distributions, the Pontryagin-Witt equation for mean exit times, and other Feynman-Kactype equations).

This research is joint with Seunggil Choi, Richard Sowers, and Lalit Vedula.