Stochastic completeness and gradient representations for sub-Riemannian manifolds
by Erlend Grong and Anton Thalmaier
Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup Pt, we give two different stochastic representations of dPtf 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.
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