Tube estimates for diffusion processes under a weak H\"ormander condition

We consider a diffusion process under a local weak H\"{o}rmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path), explicitly depending on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the non-isotropic structure of the problem, meaning that the diffusion propagates in $\mathbb{R}^2$ with different speeds in the directions $\sigma$ and $[\sigma,b]$. We establish a connection between this norm and the standard control distance.