Limits of Random Differential Equations on Manifolds

TL;DR

This paper investigates the asymptotic behavior of a family of random differential equations on manifolds driven by diffusion processes, establishing weak convergence and describing the limiting process as the parameter approaches zero.

Contribution

It provides a rigorous analysis of the limits of random differential equations on manifolds driven by fast diffusion processes, including convergence results and rate bounds.

Findings

Weak convergence of processes as epsilon approaches zero

Description of the limiting stochastic process

Upper bounds for convergence rate

Abstract

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