Stochastic completeness and gradient representations for sub-Riemannian manifolds

16 Aug 2017  ·  Grong Erlend, Thalmaier Anton ·

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. read more

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Differential Geometry Probability