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 ${\sigma }_1,\dots ,{\sigma }_d$ 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 $\delta$, 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.

References in corpus (1)

• On gradient bounds for the heat kernel on the Heisenberg group