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Degenerate diffusions and harmonic analysis on \(\mathrm{SE}(3)\): a tutorial. (English) Zbl 1391.43003

Albeverio, Sergio (ed.) et al., Stochastic geometric mechanics. CIB, Lausanne, Switzerland, January–June 2015. Cham: Springer (ISBN 978-3-319-63452-4/hbk; 978-3-319-63453-1/ebook). Springer Proceedings in Mathematics & Statistics 202, 77-99 (2017).
Summary: The group of special Euclidean transformations, \(\mathrm{SE}(3)\), describes rigid-body motion in three-dimensional Euclidean space. This group plays an important role in the fields of geometric mechanics, control, and robotics because it is the configuration space for a rigid body. This tutorial reviews the properties of this group, and explains how stochastic differential equations which lead to degenerate diffusion processes on \(\mathrm{SE}(3)\) arise in the context of applications in state estimation and in molecular dynamics. Representation theory and harmonic analysis on this group are reviewed, and it is shown how these pure mathematical methods can be used as computational tools for computing quantities of interest in applications.
For the entire collection see [Zbl 1386.70002].

MSC:

43A10 Measure algebras on groups, semigroups, etc.
22E40 Discrete subgroups of Lie groups
70E15 Free motion of a rigid body
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