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We study lower and upper bounds for the probability that a diffusion processin $\mathbb{R}^n$ remains in a tube around a skeleton path up to a fixed time.We assume that the diffusion coefficients $\sigma_1,\ldots,\sigma_d$ maydegenerate but they satisfy a strong H\"ormander condition involving the firstorder Lie brackets around the skeleton of interest. The tube is written interms 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 techniquewhich strongly uses the lower and upper bounds for the density proved in thepart I.

Diffusions under a local strong H\"ormander condition. Part II: tube estimates