Stochastic completeness and gradient representations for sub-Riemannian manifolds

Abstract

Given a second order partial differential operator $L$ satisfying the strong Hörmander condition with corresponding heat semigroup $P_t$, we give two different stochastic representations of $d{P}_{t}f$ 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.

34 pages