An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements also hold in the case of a Lie group. By using the wrapping map of Dooley and ildberger, we show how to wrap a Brownian motion to a compact Lie group from its Lie algebra (viewed as a Euclidean space) and find the heat kernel. This is achieved by considering It??o type stochastic differential equations and applying the Feynman-Ka??c theorem. We also consider wrapping Brownian motion to various symmetric spaces, where a global generalisation of Rouvi`ere???s formula and the e-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.
| Identifer | oai:union.ndltd.org:ADTP/215498 |
| Date | January 2006 |
| Creators | Maher, David Graham, School of Mathematics, UNSW |
| Publisher | Awarded by:University of New South Wales. School of Mathematics |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Copyright David Graham Maher, http://unsworks.unsw.edu.au/copyright |
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