# Perturbation of Conservation Laws and Averaging on Manifolds

## TL;DR

This paper establishes a stochastic averaging theorem for systems with interacting slow and fast variables, analyzing the behavior of invariant measures and applying conservation laws to derive slow-fast stochastic differential equations.

## Contribution

It introduces a new averaging theorem for stochastic systems with conservation laws, extending classical results to more general operators and manifold settings.

## Key findings

- Proves a locally uniform law of large numbers for Markov processes with generator ${\mathcal L}_x$
- Shows continuous dependence of invariant measures on parameters
- Demonstrates applications to physics and geometry through nonlinear conservation laws

## Abstract

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.