Small stochastic perturbations of Hamiltonian flows: a PDE approach

We present a PDE approach to the study of averaging principles for (small) stochastic perturbations of Hamiltonian flows in 2D, which is based on a recent joint work with Takis Souganidis. Such problems were introduced by Freidlin and Wentzel and have been the subject of extensive study in the last decade. If the Hamiltonian flow has critical points, then the averaging principle exhibits complicated behavior. Asymptotically, the slow (averaged) motion has 1D character and takes place on a graph, and the question is to identify the limit motion in terms of PDE problems. In their original work Freidlin and Wentzell, using probabilistic techniques, considered perturbations by Brownian motions, while later Freidlin and Weber studied, combining probabilistic and analytic techniques based on hypoelliptic operators, a special degenerate case. Recently Sowers revisited the uniformly elliptic case and constructed what amounts to approximate correctors for the averaging problem. Our approach is based on PDE techniques and is applied to general degenerate elliptic operators.