Tubes estimates for diffusion processes under a local H\"ormander condition of order one

TL;DR

This paper provides lower bounds for the probability that a diffusion process stays close to a skeleton curve under a local Hörmander condition of order one, using a non-isotropic distance reflecting the process's directional speeds.

Contribution

It establishes lower bounds for diffusion process tube estimates under a local Hörmander condition of order one, connecting a non-isotropic distance with the standard control metric.

Findings

The non-isotropic distance d is locally equivalent to the control metric d_c.

The probability bounds hold for the distance d and also for d_c.

The results apply under the Hörmander condition involving diffusion vector fields and their Lie brackets.

Abstract

We consider a diffusion process $X_{t}$ and a skeleton curve $x_{t}(\phi)$ and we give a lower bound for $P(\sup_{t\leq T}d(X_{t},x_{t}(\phi))\leq R)$. This result is obtained under the hypothesis that the strong H\"{o}rmander condition of order one (which involves the diffusion vector fields and the first Lie brackets) holds in every point $x_{t}(\phi),0\leq t\leq T.$ Here $d$ is a distance which reflects the non isotropic behavior of the diffusion process which moves with speed $\sqrt{t}$ in the directions of the diffusion vector fields but with speed $t$ in the directions of the first order Lie brackets. We prove that $d$ is locally equivalent with the standard control metric $d_{c}$ and that our estimates hold for $d_{c}$ as well.