Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds Potential Anal https://doi.org/10.1007/s11118-018-9710-x Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds 1,2 3 Erlend Grong · Anton Thalmaier Received: 16 August 2017 / Accepted: 21 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Given a second order partial differential operator L satisfying the strong Hormander ¨ condition with corresponding heat semigroup P , we give two different stochas- tic representations of dP f for a bounded smooth function f . We show that the first identity can be used to prove infinite lifetime of a diffusion of 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. Keywords Diffusion process · Stochastic completeness · Hypoelliptic operators · Gradient bound · Sub-Riemannian geometry Mathematics Subject Classification (2010) 60D05 · 35P99 · 53C17 · 47B25 This work has been supported by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746). The first author supported by project 249980/F20 of the Norwegian Research Council. Erlend Grong erlend.grong@math.uib.no Anton Thalmaier anton.thalmaier@uni.lu Universite http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Potential Analysis Springer Journals

Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

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Publisher
Springer Netherlands
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Potential Theory; Probability Theory and Stochastic Processes; Geometry; Functional Analysis
ISSN
0926-2601
eISSN
1572-929X
D.O.I.
10.1007/s11118-018-9710-x
Publisher site
See Article on Publisher Site

Abstract

Potential Anal https://doi.org/10.1007/s11118-018-9710-x Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds 1,2 3 Erlend Grong · Anton Thalmaier Received: 16 August 2017 / Accepted: 21 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract Given a second order partial differential operator L satisfying the strong Hormander ¨ condition with corresponding heat semigroup P , we give two different stochas- tic representations of dP f for a bounded smooth function f . We show that the first identity can be used to prove infinite lifetime of a diffusion of 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. Keywords Diffusion process · Stochastic completeness · Hypoelliptic operators · Gradient bound · Sub-Riemannian geometry Mathematics Subject Classification (2010) 60D05 · 35P99 · 53C17 · 47B25 This work has been supported by the Fonds National de la Recherche Luxembourg (FNR) under the OPEN scheme (project GEOMREV O14/7628746). The first author supported by project 249980/F20 of the Norwegian Research Council. Erlend Grong erlend.grong@math.uib.no Anton Thalmaier anton.thalmaier@uni.lu Universite

Journal

Potential AnalysisSpringer Journals

Published: May 28, 2018

References

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