Stochastic Averaging for Randomly Perturbed Hamiltonian Systems

The technique of averaging (deterministic or stochastic) is used to effect model reduction in systems which possess dynamics on multiple time scales. Roughly speaking, the idea is to obtain reduced models for slowly varying quantities by taking long-term averages in rapidly varying quantities. In this talk, we overview some problems in stochastic averaging for a class of planar Hamiltonian systems subjected to small white noise perturbations. Under a suitable time-rescaling, the dynamics can be seen to comprise a fast rotation along orbits of the Hamiltonian flow together  with a slow transversal diffusion across orbits. As the ratio of speeds of the slow to fast variables goes to zero, the limiting motion is given by a graph-valued Markov process governed by linear second-order differential operators on the legs of the graph together with so-called glueing conditions at each vertex. We present some recent results for the case of a planar Hamiltonian system with skew random perturbations.