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Stochastic completeness and gradient representations for sub-Riemannian manifolds

by Erlend Grong and Anton Thalmaier

**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 *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 *½L*, while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot groups. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry.

The paper is available here:

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