Random Perturbations of Hamiltonian Systems

Summary.

In Chaps. 8–9 the situation is considered where the averaging is due to the mixing in the non-perturbed dynamical system, which is supposed to be a Hamiltonian one. In all averaging-principle problems we can single out in the perturbed system fast components and slow ones. Smallness of the perturbations must be considered as compared to the speed of the motion according to the dynamical system; so one possible model of small perturbations is considering the dynamical-system motion with velocities multiplied by a large parameter, the slow motion due to the perturbations being not too fast and not too slow. The fast motion is approximately the same as the sped-up non-perturbed motion—non-random, in fact; but the slow motion, considered in another time scale, remains random even when the randomness of the perturbation goes to zero. The question arises on what space should we consider this slow motion. If the non-perturbed system has no saddle-type points, we can, in the one-degree-of-freedom case, characterize the slow motion by the value of the Hamiltonian (in the multidimensional case with several first integrals, by the values of those first integrals). If there are saddle-type points, the slow motion should be considered on a graph in the one-degree-of-freedom case, and on an “open-book” type space in the multidimensional case. In the case of there being saddle-type points the randomness of the slow-motion component does not disappear as the diffusion coefficient of the perturbation goes to zero; and the limiting random process does not depend on the choice of the diffusion coefficients, that is on the truly random part of the perturbations—so it can be considered as the intrinsic property of the dynamical system with small non-random perturbations, due to high instability at the saddle-type points. The technique of proofs relies on martingale problems.

The one-degree-of-freedom case is considered in Chap. 8. In this case the slow motion is considered on a graph, and partial differential equations on it are reduced to several ordinary differential equations, whose solutions depend on a finite number of constants that is equal to the number of gluing conditions imposed on the solution. This takes care of the problem of existence and uniqueness of the solution of the martingale problem on the graph.