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 $\delta$. We assume that the diffusion coefficients ${\sigma }_1,\dots ,{\sigma }_d$ 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 $\delta$, 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 $[{\sigma }_i,{\sigma }_j]$. 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.

References in corpus (2)

• On gradient bounds for the heat kernel on the Heisenberg group
• Diffusions under a local strong Hörmander condition. Part II: tube estimates