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

Abstract

We consider a diffusion process $X_t$ and a skeleton curve $x_t(\varphi )$ and we give a lower bound for $P({\sup }_{t\le T}d(X_t,x_t(\varphi ))\le R)$. 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 $x_t(\varphi ),0\le t\le T.$ Here $d$ is a distance which reflects the non isotropic behavior of the diffusion process which moves with speed $\sqrt{t}$ 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.