Tubes estimates for diffusion processes under a local Hörmander condition of order one

We consider a diffusion process X t and a skeleton curve x t (φ) and we give a lower bound for P (sup t≤T d(X t , x t (φ)) ≤ R). This result is obtained under the hypothesis that the strong Hörmander condition of order one (which involves the diffusion vector fields and the first Lie brackets) holds in every point x t (φ), 0 ≤ t ≤ T. Here d is a distance which reflects the non isotropic behavior of the diffusion process which moves with speed √ t in the directions of the diffusion vector fields but with speed t in the directions of the first order Lie brackets. We prove that d is locally equivalent with the standard control metric d c and that our estimates hold for d c as well