Diffusions under a local strong Hörmander condition. Part II: tube estimates

Abstract

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 $σ_1,\ldots,σ_d$ may degenerate but they satisfy a strong Hörmander 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 $δ$, the diffusion process propagates with speed $\sqrtδ$ in the direction of the diffusion vector fields $σ_{j}$ and with speed $δ=\sqrtδ\times \sqrtδ$ in the direction of $[σ_{i},σ_{j}]$. The proof consists in a concatenation technique which strongly uses the lower and upper bounds for the density proved in the part I.