Averaging of incompressible flows on two-dimensional surfaces

Abstract

We consider a process on a compact two-dimensional surface which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell that if the unperturbed motion is periodic for almost all the initial points, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation when the unperturbed motion is not periodic and the flow has ergodic components of positive measure. We show that the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertices corresponding to the ergodic components of the flow. As shown in a companion paper (Deterministic and stochastic perturbations of area preserving flows on a two-dimensional torus, by D. Dolgopyat, M. Freidlin, and L. Koralov), these results allow one to describe the viscosity regularization of dissipative deterministic perturbations of incompressible flows. In the case of surfaces of higher genus, the limiting process may exhibit intermittent behavior in the sense that the time axis can be divided into the intervals of random lengths where the particle stays inside one ergodic component separated by the intervals where the particle moves deterministically between the components.