An averaging principle for diffusions in foliated spaces

TL;DR

This paper establishes an averaging principle for stochastic differential equations on foliated manifolds, showing that small transversal perturbations lead to convergence towards a deterministic ODE, with quantified convergence rates.

Contribution

It generalizes previous results by extending the averaging principle to diffusions on foliated spaces, including integrable stochastic Hamiltonian systems.

Findings

Transversal component converges to a deterministic ODE solution.

Convergence rate of the averaging process is estimated.

Results apply to a broad class of foliated stochastic systems.

Abstract

Consider an SDE on a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behavior of a small transversal perturbation of order $\varepsilon$. An average principle is shown to hold such that the component transversal to the leaves converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to invariant measures on the leaves, as $\varepsilon$ goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system.