Perturbation of Conservation Laws and Averaging on Manifolds

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 Hörmander'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.

48 pages

References in corpus (4)

• An averaging principle for a completely integrable stochastic Hamiltonian system
• Small Mass Limit of a Langevin Equation on a Manifold
• Pointwise Gradient Bounds for Degenerate Semigroups (of UFG type)
• Limits of Random Differential Equations on Manifolds

Cited by in corpus (2)

• Averaging dynamics driven by fractional Brownian motion
• Slow-Fast Systems with Fractional Environment and Dynamics