Diffusions under a local strong Hörmander condition. Part I: density estimates

Abstract

We study lower and upper bounds for the density of a diffusion process in ${\mathbb{R}}^n$ in a small (but not asymptotic) time, say $δ$. We assume that the diffusion coefficients $σ_1,\ldots,σ_d$ may degenerate at the starting time $0$ and point $x_0$ but they satisfy a strong Hörmander condition involving the first order Lie brackets. The density estimates are 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 $[σ_{i},σ_{j}]$. In the second part of this paper, such estimates will be used in order to study lower and upper bounds for the probability that the diffusion process remains in a tube around a skeleton path up to a fixed time.