Averaging along diffusions in foliated manifolds

Consider a diffusion in a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behaviour of a small transversal perturbation of order $\epsilon$ which destroys the foliated trajectories. An average principle is shown to hold such that the transversal component 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 of the original foliated system, as $\epsilon$ 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. The diffusion can be generated either by Stratonovich SDE, Lévy noise (Marcus equation) and others. These results, with different types of diffusions, have been obtained in colaborations with Michael Högele, I. Gonzales-Gargate and P. H. da Costa.