Stochastic completeness and gradient representations for sub-Riemannian manifolds

TL;DR

This paper explores stochastic representations of the heat semigroup's gradient on sub-Riemannian manifolds, providing tools for analyzing diffusion processes and gradient bounds in these geometries.

Contribution

It introduces two stochastic formulas for the gradient of the heat semigroup on sub-Riemannian manifolds, linking geometry with stochastic analysis.

Findings

Proves infinite lifetime of certain diffusions.

Provides explicit pointwise gradient bounds on Carnot groups.

Establishes connections between sub-Riemannian geometry and stochastic processes.

Abstract

Given a second order partial differential operator $L$ satisfying the strong H\"ormander 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.