Perturbation of Conservation Laws and Averaging on Manifolds

We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator ${\mathcal L}_x$ for which we obtain a locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter $x$. These results are obtained under the assumption that ${\mathcal L}_x$ satisfies H\"ormander's bracket conditions, or more generally ${\mathcal L}_x$ is a family of Fredholm operators with sub-elliptic estimates. On the other hand a conservation law of a dynamical system can be used as a tool for separating the scales in singular perturbation problems. We also study a number of motivating examples from mathematical physics and from geometry where we use non-linear conservation laws to deduce slow-fast systems of stochastic differential equations.