Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere) (Q2759046)

A small random perturbation of a 2-dimensional Hamiltonian ordinary differential equation by white noise is investigated by the method of stochastic averaging. The novelty here is that the Hamiltonian is not assumed to be non-degenerate at its critical points, but rather the set of critical points is allowed to have a non-empty interior. Averaging does therefore not affect the behaviour of the system in the interior of this set. The result of the paper says that averaging gives a Markov process on a stratified space, which can be described as a sphere with a line attached. Glueing conditions at the junction of the sphere and the line have to be identified to describe the behaviour of the Markov process. The methods used may be considered classical, but they employ fairly sophisticated investigations of the limiting behaviour of the generator of the Markov process for the intensity of the perturbation tending to zero. The difficulties arise, in particular, near the boundary of the set of critical points of the Hamiltonian, where the analysis turns into a singular perturbation problem. The investigation is performed in a very careful and well structured manner; the paper is very well written.