Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

Abstract

Given a second order partial differential operator L satisfying the strong Hormander condition with corresponding heat semigroup $P_{t}$ , we give two different stochastic representations of $dP_{t} f$ for a bounded smooth function f. We show that the first identity can be used to prove infinite lifetime of a diffusion of $\frac {1}{2} L$ , while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot group. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry.