  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.
