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Erschienen in:
2012 | OriginalPaper | Buchkapitel
Stochastic Processes on Lie Groups
verfasst von : Gregory S. Chirikjian
Erschienen in: Stochastic Models, Information Theory, and Lie Groups, Volume 2
Verlag: Birkhäuser Boston
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As has been discussed in earlier chapters, it is possible to define probability densities on Lie groups and to compute convolutions. Since Lie groups are by definition also analytic manifolds, the methodology from Chapter 8 of Volume 1 can be used to define SDEs and Fokker–Planck equations. However, the added structure provided by Lie groups means that these equations can be derived in completely Lie-theoretic terms without ever referring to coordinates or charts. In addition, the natural embedding of Lie groups into matrices means that SDEs can be written extrinsically as well. These topics are discussed here, along with related topics from the field of probability and statistics on Lie groups. These include answering the questions: “How can the concepts of mean and covariance of a pdf on a Lie group be defined?” “If I only care how the mean and covariance behave as a function of time, can I obtain these without solving the Fokker–Planck equation?”