Limits of Random Differential Equations on Manifolds

Abstract

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form ${\sum }_k{Y}_k{\alpha }_k(z_t^{\epsilon }(\omega ))$ where $Y_k$ are vector fields, $\epsilon$ is a positive number, $z_t^{\epsilon }$ is a $\frac{1}{\epsilon }{\mathcal{L}}_0$ diffusion process taking values in possibly a different manifold, ${\alpha }_k$ are annihilators of $ker({\mathcal{L}}_0^{\ast })$. Under Hörmander type conditions on ${\mathcal{L}}_0$ we prove that, as $\epsilon$ approaches zero, the stochastic processes $y_{\frac{t}{\epsilon }}^{\epsilon }$ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.

46 pages, To appear in Probability Theory and Related Fields In this version, we add a note in proof for the published version

References in corpus (3)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• An averaging principle for a completely integrable stochastic Hamiltonian system
• Effective Diffusions with Intertwined Structures

Cited by in corpus (2)

• Homogenisation On Homogeneous Spaces
• Perturbation of Conservation Laws and Averaging on Manifolds