Chapter

Operators and measures

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion

Published in print August 2012 | ISBN: 9780199654109
Published online January 2013 | e-ISBN: 9780191745676 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199654109.003.0003

Series: Oxford Mathematical Monographs

Operators and measures

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This chapter deals with operators and measures. The Casimir operator Ξ on PSO(1, d), i.e., the second-order differential operator associated with the Killing form, is the fundamental operator of the theory. It induces the Laplace operator D on the affine group Ad, and the hyperbolic Laplacian ∆. After proving some fundamental properties of Haar measures on groups, we determine the Haar measure of PSO(1, d). The chapter ends with a presentation of harmonic, Liouville and volume measures. These can all be derived from the Haar measure, and their analytical expressions are derived in this way.

Keywords: Casimir operator; Laplace operators; Haar measure; harmonic measures; Liouville measure

Chapter.  16871 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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