Tube estimates for diffusion processes under a weak H\"ormander condition

TL;DR

This paper derives Gaussian estimates and probability bounds for diffusion processes satisfying a weak H"ormander condition, highlighting the non-isotropic structure and connecting it to control distance.

Contribution

It provides new Gaussian estimates and tube probability bounds for diffusions under a local weak H"ormander condition, considering non-isotropic propagation speeds.

Findings

Gaussian density estimates in short time

Exponential bounds for tube probabilities

Connection between non-isotropic norm and control distance

Abstract

We consider a diffusion process under a local weak H\"{o}rmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path), explicitly depending on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the non-isotropic structure of the problem, meaning that the diffusion propagates in $\mathbb{R}^2$ with different speeds in the directions $\sigma$ and $[\sigma,b]$. We establish a connection between this norm and the standard control distance.