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Generalised Weitzenböck formulae for differential operators in Hörmander form. (English) Zbl 1405.58016

Eberle, Andreas (ed.) et al., Stochastic partial differential equations and related fields. In honor of Michael Röckner, SPDERF, Bielefeld, Germany, October 10–14, 2016. Cham: Springer (ISBN 978-3-319-74928-0/hbk; 978-3-319-74929-7/ebook). Springer Proceedings in Mathematics & Statistics 229, 319-331 (2018).
Summary: The decomposition of a class of diffusion processes, due to Elworthy-LeJan-Li is described. In particular it applies to processes such as derivative processes coming from stochastic flows. How this decomposition leads automatically to Weitzenböck type formula for related operators acting on sections of associated vector bundles is described in detail clarifying the difference between the action of flows on vector fields and on forms noted recently by Shizan Fang and Dejun Luo. Remarks are made on the possible application to higher order derivative formulae and estimates for heat semigroups, and also to certain diffusions with sub-Riemannian generators using Baudoin’s generalised Levi-Civita semi-connection.
For the entire collection see [Zbl 1402.35005].

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60G35 Signal detection and filtering (aspects of stochastic processes)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J60 Diffusion processes
93E11 Filtering in stochastic control theory
53C17 Sub-Riemannian geometry
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