Generalised Weitzenböck Formulae for Differential Operators in Hörmander Form
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.
KeywordsStochastic analysis Stochastic flows Weitzenböck formulae Diffusion of tensors Degenerate diffusions Semi-group domination
Mathematics Subject Classification58J65 (60G35 60H30 60J60 93E11 53C17)
This article is based on the cited joint work with Yves Le Jan and Xue-Mei Li. It was stimulated by correspondence and discussions with Shizan Fang and by his work with Dejun Luo, and Sect. 5 came out of correspondence with Fabrice Baudoin.
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