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 Lx for which we obtain a quantitative 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 Lx satisfies Hörmander’s bracket conditions, or more generally Lx is a family of Fredholm operators with sub-elliptic estimates. For stochastic systems in which the slow and the fast variable are not separate, conservation laws are essential ingredients for separating the scales in singular perturbation problems we demonstrate this by a number of motivating examples, from mathematical physics and from geometry, where conservation laws taking values in non-linear spaces are used to deduce slow-fast systems of stochastic differential equations