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Cartan connections for stochastic developments on sub-Riemannian manifolds
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Beschastnyi, Ivan, Habermann, Karen and Medvedev, Alexandr (2022) Cartan connections for stochastic developments on sub-Riemannian manifolds. The Journal of Geometric Analysis, 32 . 13. doi:10.1007/s12220-021-00743-9 ISSN 1050-6926.
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Official URL: http://dx.doi.org/10.1007/s12220-021-00743-9
Abstract
Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. We further provide a necessary and sufficient condition for the existence of a Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp’s volume. We illustrate the construction of a suitable Cartan connection for free sub-Riemannian structures with two generators and we discuss an example where the condition is not satisfied.
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | |||||||||
Library of Congress Subject Headings (LCSH): | Riemannian manifolds, Brownian motion processes, Stochastic processes, Connections (Mathematics) | |||||||||
Journal or Publication Title: | The Journal of Geometric Analysis | |||||||||
Publisher: | Springer New York LLC | |||||||||
ISSN: | 1050-6926 | |||||||||
Official Date: | 2022 | |||||||||
Dates: |
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Volume: | 32 | |||||||||
Article Number: | 13 | |||||||||
DOI: | 10.1007/s12220-021-00743-9 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Date of first compliant deposit: | 22 February 2022 | |||||||||
Date of first compliant Open Access: | 8 December 2022 | |||||||||
RIOXX Funder/Project Grant: |
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