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Stochastic Processes on Lie Groups

  • Gregory S. Chirikjian
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

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?”

Keywords

Brownian Motion Central Limit Theorem Planck Equation Stochastic Trajectory Noncompact Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

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