  We study lower and upper bounds for the probability that a diffusion process
in $\mathbb{R}^n$ remains in a tube around a skeleton path up to a fixed time.
We assume that the diffusion coefficients $\sigma_1,\ldots,\sigma_d$ may
degenerate but they satisfy a strong H\"ormander condition involving the first
order Lie brackets around the skeleton of interest. The tube is written in
terms of a norm which accounts for the non-isotropic structure of the problem:
in a small time $\delta$, the diffusion process propagates with speed
$\sqrt{\delta}$ in the direction of the diffusion vector fields $\sigma_{j}$
and with speed $\delta=\sqrt{\delta}\times \sqrt{\delta}$ in the direction of
$[\sigma_{i},\sigma_{j}]$. The proof consists in a concatenation technique
which strongly uses the lower and upper bounds for the density proved in the
part I.
